§1.1. Real numbers
The first acquaintance with real numbers occurs in the school course of mathematics. Any real number is represented by a finite or infinite decimal fraction.
Real (real) numbers are divided into two classes: the class of rational and the class of irrational numbers. Rational numbers are called that look like , where m and n are coprime integers, but 
. (The set of rational numbers is denoted by the letter Q). The rest of the real numbers are called irrational. Rational numbers are represented by a finite or infinite periodic fraction (the same as ordinary fractions), then those and only those real numbers that can be represented by infinite non-periodic fractions will be irrational.
For example, number 
- rational and 
, 
, 
etc. are irrational numbers.
Real numbers can also be divided into algebraic ones - the roots of a polynomial with rational coefficients (these include, in particular, all rational numbers - the roots of the equation 
) - and transcendental - all the rest (for example, numbers 
other).
The sets of all natural, integer, real numbers are denoted respectively as follows: NZ, R
(the initial letters of the words Naturel, Zahl, Reel).
§1.2. Image of real numbers on the number line. Intervals
Geometrically (for clarity), real numbers are represented by points on an infinite (in both directions) straight line, called numerical
axis. For this purpose, a point is taken on the line under consideration (the reference point is point 0), a positive direction is indicated, depicted by an arrow (usually to the right), and a scale unit is chosen, which is set aside indefinitely on both sides of the point 0. This is how integers are displayed. To depict a number with one decimal place, each segment must be divided into ten parts, and so on. Thus, each real number is represented by a point on the number line. Conversely, every point 
corresponds to a real number equal to the length of the segment 
and taken with the sign "+" or "-", depending on whether the point lies to the right or to the left of the origin. Thus, a one-to-one correspondence is established between the set of all real numbers and the set of all points of the numerical axis. The terms "real number" and "point of the numerical axis" are used as synonyms.
Symbol 
we will denote both a real number and a point corresponding to it. Positive numbers are located to the right of the point 0, negative - to the left. If a 
, then on the real axis the point 
lies to the left of the point 
. Let the point 
corresponds to a number, then the number is called the coordinate of the point, they write 
; more often, the point itself is denoted by the same letter as the number. Point 0 is the origin of coordinates. The axis is also denoted by the letter 
(fig.1.1).
Rice. 1.1. Numeric axis.
The set of all numbers lying between given numbers and is called an interval or gap; the ends may or may not belong to him. Let's clarify this. Let be 
. The set of numbers that satisfy the condition 
, is called an interval (in the narrow sense) or an open interval, denoted by the symbol 
(fig.1.2).
Rice. 1.2. Interval
A collection of numbers such that 
is called a closed interval (segment, segment) and is denoted by 
; on the numerical axis is marked as follows:
Rice. 1.3. closed interval
It differs from an open gap only in two points (ends) and . But this difference is fundamental, essential, as we will see later, for example, when studying the properties of functions.
Omitting the words "the set of all numbers (points) x such that ", etc., we note further:
and 
, denoted 
and 
half-open, or half-closed, intervals (sometimes: half-intervals);
or 
means: 
or 
and denoted 
or 
;
or 
means 
or 
and denoted 
or 
;
, denoted 
the set of all real numbers. Badges 
symbols of "infinity"; they are called improper or ideal numbers. 
§1.3. Absolute value (or modulus) of a real number
Definition. Absolute value (or module) number is called the number itself, if 
or 
if 
. The absolute value is denoted by the symbol 
. So, 
For example, 
, 
, 
.
Geometrically means point distance a to the origin of coordinates. If we have two points and , then the distance between them can be represented as 
(or 
). For example, 
that distance 
.
Properties of absolute values.
1. It follows from the definition that 
, 
, i.e 
.
2. The absolute value of the sum and difference does not exceed the sum of the absolute values: 
.
1) If 
, then 
. 2) If 
, then . ▲
3. 
.
, then by property 2: 
, i.e. 
. Similarly, if we imagine 
, then we arrive at the inequality 
▲
4. 
– follows from the definition: consider cases 
and 
.
5. 
, provided that 
The same follows from the definition.
6. Inequality 
,
, means 
. This inequality is satisfied by the points that lie between 
and 
.
7. Inequality 
is equivalent to the inequality 
, i.e. . It is an interval centered at the point of length 
. It is called 
neighborhood of a point (number) . If a 
, then the neighborhood is called punctured: this or 
. (Fig.1.4).
8. 
whence it follows that the inequality 
(
) is equivalent to the inequality 
or 
; and inequality 
determines the set of points for which 
, i.e. are points outside the segment 
, exactly: 
and 
.
§1.4. Some concepts, designations
Let us give some widely used concepts, notations from set theory, mathematical logic and other branches of modern mathematics.
1 . concept sets is one of the basic in mathematics, initial, universal - and therefore cannot be defined. It can only be described (replaced by synonyms): it is a collection, a collection of some objects, things, united by some signs. These objects are called elements sets. Examples: many grains of sand on the shore, stars in the universe, students in the classroom, the roots of the equation, points of the segment. Sets whose elements are numbers are called numerical sets. For some standard sets, special notation is introduced, for example, N,Z,R- see § 1.1.
Let be A- set and x is its element, then we write: 
; reads " x belongs A» ( 
inclusion sign for elements). If the object x not included in A, then they write 
; reads: " x not belong A". For example, 
N; 8,51
N; but 8.51 
R.
If a x is a general designation for elements of a set A, then they write 
. If it is possible to write out the designation of all elements, then write 
, 
etc. A set that does not contain a single element is called an empty set and is denoted by the symbol ; for example, the set of (real) roots of the equation 
there is an empty one.
The set is called final if it consists of a finite number of elements. If, however, no matter what natural number N is taken, in the set A there are more elements than N, then A called endless set: there are infinitely many elements in it.
If every element of the set ^A belongs to the set B, then 
called a part or subset of a set B and write 
; reads " A contained in B» ( 
there is an inclusion sign for sets). For example, NZR. If 
, then we say that the sets A and B equal and write 
. Otherwise, write 
. For example, if 
, a 
set of roots of the equation 
, then .
The set of elements of both sets A and B called association sets and is denoted 
(sometimes 
). The set of elements belonging to and A and B, is called intersection sets and is denoted 
. The set of all elements of the set ^A, which are not included in B, is called difference sets and is denoted 
. Schematically, these operations can be depicted as follows:
If a one-to-one correspondence can be established between the elements of the sets, then they say that these sets are equivalent and write 
. Any set A, equivalent to the set of natural numbers N= called countable or countable. In other words, a set is called countable if its elements can be numbered, placed in an infinite subsequence 
, all members of which are different: 
at 
, and it can be written as . Other infinite sets are called uncountable. Countable, except for the set itself N, there will be, for example, sets 
, Z. It turns out that the sets of all rational and algebraic numbers are countable, and the equivalent sets of all irrational, transcendental, real numbers and points of any interval are uncountable. They say that the latter have the power of the continuum (the power is a generalization of the concept of the number (number) of elements for an infinite set).
2
. Let there be two statements, two facts: and 
. Symbol 
means: "if true, then true and" or "follows", "implies there is a root of the equation has a property from English Exist- exist.
Recording: 
, or 
, means: there is (at least one) object that has the property 
. A record 
, or 
, means: all have the property . In particular, we can write: 
and .
The concepts of "set", "element", "membership of an element in a set" are the primary concepts of mathematics. A bunch of- any collection (aggregate) of any items .
A is a subset of B if each element of set A is an element of set B, i.e. AÌV Û (хОА u хОВ).
Two sets are equal if they consist of the same elements. We are talking about set-theoretic equality (not to be confused with equality between numbers): A=B Û AÌB Ù BÌA.
Union of two sets consists of elements belonging to at least one of the sets, i.e. xOAÈV Û xOAÚ xOV.
intersection consists of all elements simultaneously belonging to both set A and set B: хОАЗВ Û хОА u хОВ.
Difference consists of all elements of A that do not belong to B, i.e. xO A\B Û xOA ÙxPB.
Cartesian product C=A´B of sets A and B is the set of all possible pairs ( x,y), where the first element X each pair belongs to A, and its second element at belongs to V.
The subset F of the Cartesian product A´B is called mapping from set A to set B , if the condition is met: (" XОА)($! pair ( x.y)ОF). At the same time they write: A.V.
The terms "mapping" and "function" are synonymous. If ("хОА)($! уОВ): ( x,y)нF, then the element atÎ AT called way X when displaying F and write it like this: at=F( X). Element X at the same time is prototype (one of the possible) element y.
Consider set of rational numbers Q - the set of all integers and the set of all fractions (positive and negative). Every rational number can be represented as a quotient, for example, 1 =4/3=8/6=12/9=…. There are many such representations, but only one of them is irreducible. .
AT any rational number can be uniquely represented as a fraction p/q, where pОZ, qОN, the numbers p, q are coprime.
Properties of the set Q:
1. Closure with respect to arithmetic operations. The result of addition, subtraction, multiplication, raising to a natural power, division (except division by 0) of rational numbers is a rational number: ; ; 
.
2. Ordering: (" x, yОQ, x¹y)®( x
And: 1) a>b, b>c Þ a>c; 2)a -b.
3. Density. Between any two rational numbers x, y there is a third rational number (for example, c= ):
("x, yОQ, x<y)($cОQ) : ( X
On the set Q, you can perform 4 arithmetic operations, solve systems of linear equations, but quadratic equations of the form x 2 \u003d a, a N are not always solvable in the set Q.
Theorem. There is no number хОQ, whose square is 2.
g Let there be such a fraction X=p/q, where the numbers p and q are coprime and X 2=2. Then (p/q) 2 =2. Hence,
The right side of (1) is divisible by 2, so p 2 is an even number. Thus p=2n (n-integer). Then q must be an odd number.
Returning to (1), we have 4n 2 =2q 2 . Therefore q 2 \u003d 2n 2. Similarly, we make sure that q is divisible by 2, i.e. q is an even number. By contradiction, the theorem is proved.n
geometric representation of rational numbers. Putting a single segment from the origin of coordinates 1, 2, 3 ... times to the right, we get the points of the coordinate line, which correspond to natural numbers. Putting aside similarly to the left, we get the points corresponding to negative integers. Let's take 1/q(q= 2,3,4 … ) part of a single segment and we will postpone it on both sides of the origin R once. We get points of a straight line corresponding to numbers of the form ±p/q (pОZ, qОN). If p, q run through all pairs of coprime numbers, then on the line we have all points corresponding to fractional numbers. Thus, according to the accepted method, each rational number corresponds to a single point of the coordinate line.
Is there a single rational number for every point? Is the line filled entirely with rational numbers?
It turns out that there are points on the coordinate line that do not correspond to any rational numbers. We construct an isosceles right triangle on a single segment. The point N does not correspond to a rational number, since if ON=x- rationally x 2 = 2, which cannot be.
There are infinitely many points similar to the point N on the line. Take the rational parts of the segment x=ON, those. X. If we postpone them to the right, then no rational number will correspond to each of the ends of any of these segments. Assuming that the length of the segment is expressed by a rational number x=, we get that x=- rational. This contradicts what was proved above.
Rational numbers are not enough for each point of the coordinate line to be associated with some rational number.
Let's build set of real numbers R through endless decimals.
According to the “corner” division algorithm, any rational number can be represented as a finite or infinite periodic decimal fraction. When the denominator of p/q has no prime divisors other than 2 and 5, i.e. q=2 m ×5 k , then the result will be the final decimal fraction p/q=a 0 ,a 1 a 2 …a n . Other fractions can only have infinite decimal expansions.
Knowing an infinite periodic decimal fraction, you can find a rational number, the representation of which it is. But any finite decimal fraction can be represented as an infinite decimal fraction in one of the following ways:
a 0 ,a 1 a 2 …a n = a 0 ,a 1 a 2 …a n 000…=a 0 ,a 1 a 2 …(a n -1)999… (2)
For example, for an infinite decimal X=0,(9) we have 10 X=9,(9). If we subtract the original number from 10x, we get 9 X=9 or 1=1,(0)=0,(9).
A one-to-one correspondence is established between the set of all rational numbers and the set of all infinite periodic decimal fractions if we identify the infinite decimal fraction with the number 9 in the period with the corresponding infinite decimal fraction with the digit 0 in the period according to the rule (2).
Let us agree to use such infinite periodic fractions that do not have the number 9 in the period. If an infinite periodic decimal fraction with the number 9 in the period arises in the process of reasoning, then we will replace it with an infinite decimal fraction with zero in the period, i.e. instead of 1,999… we will take 2,000…
Definition of an irrational number. In addition to infinite decimal periodic fractions, there are non-periodic decimal fractions. For example, 0.1010010001… or 27.1234567891011… (natural numbers follow the decimal point).
Consider an infinite decimal fraction of the form ±a 0 , a 1 a 2 …a n … (3)
This fraction is defined by specifying a “+” or “–” sign, a non-negative integer a 0 and a sequence of decimal places a 1 ,a 2 ,…,a n ,… (the set of decimal places consists of ten numbers: 0, 1, 2,…, nine).
We call any fraction of the form (3) real (real) number. If there is a “+” sign before the fraction (3), it is usually omitted and written a 0, a 1 a 2 ... a n ... (4)
A number of the form (4) will be called non-negative real number, and in the case when at least one of the numbers a 0 , a 1 , a 2 , …, a n is different from zero, – positive real number. If in expression (3) the “-” sign is taken, then this is a negative number.
The union of the sets of rational and irrational numbers form the set of real numbers (QÈJ=R). If the infinite decimal fraction (3) is periodic, then this is a rational number, when the fraction is non-periodic, it is irrational.
Two non-negative real numbers a=a 0 ,a 1 a 2 …a n …, b=b 0 ,b 1 b 2 …b n …. called equal(they write a=b), if a n = b n at n=0,1,2… The number a is less than the number b(they write a<b), if either a 0 or a 0 = b 0 and there is a number m, what a k =b k (k=0,1,2,…m-1), a a m , i.e. a Û (a 0 Ú ($mОN: a k =b k (k= ), a m ). The term " a>b».
To compare arbitrary real numbers, we introduce the concept " modulus of a» . Modulus of a real number a \u003d ± a 0, a 1 a 2 ... a n ... such a non-negative real number is called, which is represented by the same infinite decimal fraction, but taken with the “+” sign, i.e. ½ a½= a 0 , a 1 a 2 …a n … and½ a½³0. If a a - non-negative, b is a negative number, then a>b. If both numbers are negative ( a<0, b<0 ), then we assume that: 1) a=b, if ½ a½ = ½ b½; 2) a , if ½ a½ > ½ b½.
Properties of the set R:
I. Order Properties:
1. For every pair of real numbers a and b there is one and only one relation: a=b, a
2. If a
3. If a , then there is a number c such that a< с .
II. Properties of addition and subtraction operations:
4. a+b=b+a(commutativity).
5. (a+b)+c=a+(b+c) (associativity).
6. a+0=a.
7. a+(-a)= 0.
8. out a Þ a+c
III. Properties of multiplication and division operations:
9. a×b=b×a .
10. (a×b)×c=a×(b×c).
11. a×1=a.
12. a×(1/a)=1 (a¹0).
13. (a + b) × c \u003d ac + bc(distributivity).
14. if a and c>0, then a×s
IV. Archimedean property("cОR)($nОN) : (n>c).
Whatever the number сОR, there exists nОN such that n>c.
v. Continuity property of real numbers. Let two non-empty sets AÌR and BÌR be such that any element aОА will be no more ( a£ b) of any element bнB. Then Dedekind's continuity principle asserts the existence of a number c such that for all aнА and bнB the condition a£c£ b:
(" AÌR, BÌR):(" aОA, bОB ® a£b)($cОR): (" aÎA, bÎB® a£c£b).
We will identify the set R with the set of points of the real line, and call the real numbers points.
Geometrically real numbers, like rational numbers, are represented by points on a straight line.
Let be l - an arbitrary straight line, and O - some of its points (Fig. 58). Every positive real number α put in correspondence the point A, lying to the right of O at a distance of α units of length.
If, for example, α = 2.1356..., then
2 < α
< 3
2,1 < α
< 2,2
2,13 < α
< 2,14
etc. It is obvious that the point A in this case must be on the line l to the right of the points corresponding to the numbers
2; 2,1; 2,13; ... ,
but to the left of the points corresponding to the numbers
3; 2,2; 2,14; ... .
It can be shown that these conditions define on the line l the only point A, which we consider as the geometric image of a real number α = 2,1356... .
Likewise, every negative real number β put in correspondence the point B lying to the left of O at a distance of | β | units of length. Finally, we assign the point O to the number "zero".
So, the number 1 will be displayed on a straight line l point A, located to the right of O at a distance of one unit of length (Fig. 59), the number - √2 - point B, lying to the left of O at a distance of √2 units of length, etc.
Let's show how on a straight line l using a compass and straightedge, one can find points corresponding to the real numbers √2, √3, √4, √5, etc. To do this, we first of all show how to construct segments whose lengths are expressed by these numbers. Let AB be a segment taken as a unit of length (Fig. 60).
At point A, we restore a perpendicular to this segment and set aside on it the segment AC, equal to the segment AB. Then, applying the Pythagorean theorem to the right triangle ABC, we get; BC \u003d √AB 2 + AC 2 \u003d √1 + 1 \u003d √2
Therefore, the segment BC has length √2. Now let us restore the perpendicular to the segment BC at the point C and choose the point D on it so that the segment CD is equal to unit length AB. Then from the right triangle BCD we find:
ВD \u003d √BC 2 + CD 2 \u003d √2 + 1 \u003d √3
Therefore, the segment BD has length √3. Continuing the described process further, we could get segments BE, BF, ..., whose lengths are expressed by the numbers √4, √5, etc.
Now on the line l it is easy to find those points that serve as a geometric representation of the numbers √2, √3, √4, √5, etc.
Putting, for example, the segment BC to the right of the point O (Fig. 61), we get the point C, which serves as a geometric representation of the number √2. In the same way, putting the segment BD to the right of the point O, we get the point D", which is the geometric image of the number √3, etc.
However, one should not think that with the help of a compass and a ruler on a number line l one can find a point corresponding to any given real number. It has been proven, for example, that, having only a compass and a ruler at your disposal, it is impossible to construct a segment whose length is expressed by the number π = 3.14 ... . So on the number line l using such constructions, it is impossible to indicate a point corresponding to this number. Nevertheless, such a point exists.
So for every real number α it is possible to associate some well-defined point of the line l . This point will be separated from the starting point O at a distance of | α | units of length and be to the right of O if α > 0, and to the left of O if α < 0. Очевидно, что при этом двум неравным действительным числам будут соответствовать две различные точки прямой l . Indeed, let the number α corresponds to point A, and the number β - point B. Then, if α > β , then A will be to the right of B (Fig. 62, a); if α < β , then A will lie to the left of B (Fig. 62, b).
Speaking in § 37 about the geometric representation of rational numbers, we posed the question: can any point of a straight line be considered as a geometric image of some rational numbers? At that time we could not give an answer to this question; now we can answer it quite definitely. There are points on the line that serve as a geometric representation of irrational numbers (for example, √2). Therefore, not every point on a straight line represents a rational number. But in this case, another question arises: can any point of the real line be considered as a geometric image of some valid numbers? This issue has already been resolved positively.
Indeed, let A be an arbitrary point on the line l , lying to the right of O (Fig. 63).
The length of the segment OA is expressed by some positive real number α (see § 41). Therefore point A is the geometric image of the number α . Similarly, it is established that each point B, lying to the left of O, can be considered as a geometric image of a negative real number - β , where β - the length of the segment VO. Finally, the point O serves as a geometric representation of the number zero. It is clear that two distinct points of the line l cannot be the geometric image of the same real number.
For the reasons stated above, a straight line on which some point O is indicated as the "initial" point (for a given unit of length) is called number line.
Conclusion. The set of all real numbers and the set of all points of the real line are in a one-to-one correspondence.
This means that each real number corresponds to one, well-defined point of the number line, and, conversely, to each point of the number line, with such a correspondence, there corresponds one, well-defined real number.
An expressive geometric representation of the system of rational numbers can be obtained as follows.
On some straight line, the "numerical axis", we mark the segment from O to 1 (Fig. 8). This sets the length of the unit segment, which, generally speaking, can be chosen arbitrarily. Positive and negative integers are then depicted as a set of equally spaced points on the number axis, it is positive numbers that are marked to the right, and negative ones to the left of the point 0. To depict numbers with a denominator n, we divide each of the obtained segments of unit length into n equal parts; division points will represent fractions with denominator n. If we do this for the values of n corresponding to all natural numbers, then each rational number will be depicted by some point on the numerical axis. We shall agree to call these points "rational"; in general, the terms "rational number" and "rational point" will be used as synonyms.
In Chapter I, § 1, the inequality relation A was defined for any pair of rational points, it is natural to try to generalize the arithmetic inequality relation in such a way as to preserve this geometric order for the points under consideration. This is possible if we accept the following definition: we say that the rational number A smaller than the rational number B (A is greater than the number A (B>A), if the difference B-A is positive. It follows from this (for A between A and B are those that are both > A and segment (or segment) and is denoted by [A, B] (and the set of only intermediate points - interval(or gap), denoted by (A, B)).
The distance of an arbitrary point A from the origin 0, considered as a positive number, is called absolute value A and is denoted by the symbol
The concept of "absolute value" is defined as follows: if A≥0, then |A| = A; if A
|A + B|≤|A| + |B|,
which is true regardless of the signs A and B.
A fact of fundamental importance is expressed by the following proposition: rational points are everywhere dense on the number line. The meaning of this statement is that inside any interval, no matter how small it may be, there are rational points. To verify the validity of the stated statement, it is enough to take the number n so large that the interval will be less than the given interval (A, B); then at least one of the view points will be inside the given interval. So, there is no such interval on the number line (even the smallest imaginable) within which there would be no rational points. This implies a further corollary: every interval contains an infinite set of rational points. Indeed, if some interval contained only a finite number of rational points, then there would no longer be rational points inside the interval formed by two neighboring such points, and this contradicts what has just been proved.
TICKET 1
Rational numbers are numbers written as p/q, where q is natural. number and p is an integer.
Two numbers a=p1/q1 and b=p2/q2 are called equal if p1q2=p2q1, and p2q1 and a>b if p1q2
TICKET 2
Complex numbers. Complex numbers
An algebraic equation is an equation of the form: P n ( x) = 0, where P n ( x) - polynomial n- oh degree. A couple of real numbers x and at will be called ordered if it is specified which of them is considered the first and which - the second. Ordered pair notation: ( x, y). A complex number is an arbitrary ordered pair of real numbers. z = (x, y)-complex number.
x- real part z, y- imaginary part z. If a x= 0 and y= 0, then z= 0. Consider z 1 = (x 1 , y 1) and z 2 = (x 2 , y 2).
Definition 1. z 1 \u003d z 2 if x 1 \u003d x 2 and y 1 \u003d y 2.
Concepts > and< для комплексных чисел не вводятся.
Geometric representation and trigonometric form of complex numbers.
M( x, y) « z = x + iy.
½ OM½ = r =½ z½ = .(picture)
r is called the modulus of a complex number z.
j is called the argument of a complex number z. It is defined up to ± 2p n.
X= rcosj , y= rsinj.
z= x+ iy= r(cosj + i sinj) is the trigonometric form of complex numbers.
Statement 3.
= (cos + i sin),
= (cos + i sin ), then
= (cos( + ) + i sin( + )),
= (cos(-)+ i sin( - )) at ¹0.
Statement 4.
If a z=r (cosj + i sinj), then " natural n:
= (cos nj + i sin nj),
TICKET 3
Let be X-number set containing at least one number (non-empty set).
xÎ X- x contained in X. ; xÏ X- x not belong X.
Definition: A bunch of X is called bounded from above (from below) if there exists a number M(m) such that for any x Î X the inequality x £ M (x ³ m), while the number M is called the upper (lower) bound of the set X. A bunch of X is called bounded from above if $ M, " x Î X: x £ M. Definition set unbounded from above. A bunch of X is called unbounded from above if " M $ x Î X: x> M Definition a bunch of X is called bounded if it is bounded above and below, i.e. $ M, m such that " x Î X: m £ x £ M. Equivalent definition of limited set: Set X is called bounded if $ A > 0, " x Î X: ½ x½£ A. Definition: The least of the upper bounds of a set bounded above X is called its least upper bound, and is denoted Sup X
(supremum). =Sup X. Similarly, one can determine the exact
bottom edge. equivalent definition exact top edge:
The number is called the least upper bound of the set X, if: 1) " x Î X: X£ (this condition shows that is one of the upper bounds). 2) " < $ x Î X: X> (this condition shows that -
the smallest of the upper bounds).
sup X= :
1. " xÎ X: x £ .
2. " < $ xÎ X: x> .
inf X(infimum) is the least infimum. Let us pose the question: does every bounded set have exact faces?
Example: X= {x: x>0) has no smallest number.
Theorem on the existence of an exact upper (lower) face. Any non-empty upper (lower) bounding set xОR has a point upper (lower) bound.
The theorem on the separability of numerical plurals:▀▀▄
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If each number n (n=1,2,3..) is assigned a certain number Xn, then they say that it is defined and given subsequence x1, x2 …, write (Xn), (Xn). Example: Xn=(-1)^n: -1,1,-1,1,… from above (from below) if the number of points x=x1,x2,…xn lying on the real axis is limited from above (from below), i. $C:Xn£C" Last limit: the number a is called the limit of the last if for any ε>0 $ : N (N=N/(ε)). "n>N the inequality |Xn-a|<ε. Т.е. – ε
at n > N.
Uniqueness of the limit bounded and convergent sequence
Property 1: A convergent sequence has only one limit.
Proof: by contradiction let a and b limits of a convergent sequence (x n ), where a is not equal to b. consider infinitesimal sequences (α n )=(x n -a) and (β n )=(x n -b). Because all elements of b.m. sequences (α n -β n ) have the same value b-a, then by property b.m. sequences b-a=0 i.e. b=a and we have come to a contradiction.
Property 2: The convergent sequence is bounded.
Proof: Let a be the limit of a convergent sequence (x n ), then α n =x n -a is an element of b.m. sequences. Take any ε>0 and use it to find N ε: / x n -a/< ε при n>Nε . Denote by b the largest of the numbers ε+/а/, /х1/, /х2/,…,/х N ε-1 /,х N ε . It is obvious that / x n /
Note: a bounded sequence may or may not be convergent.
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The sequence a n is called infinitesimal, which means that the limit of this sequence after is 0.
a n is an infinitesimal Û lim(n ® + ¥)a n =0 i.e. for any ε>0 there exists N such that |a n |<ε
Theorem. The sum of the infinitesimal is the infinitesimal.
a n b n ®infinitely small Þ a n +b n is infinitely small.
Proof.
a n - infinitesimal Û "ε>0 $ N 1:" n >N 1 z |a n |<ε
b n - infinitesimal Û "ε>0 $ N 2:" n >N 2 z |b n |<ε
Let us set N=max(N 1 ,N 2 ), then for any n>N z both inequalities hold simultaneously:
|a n |<ε |a n +b n |£|a n |+|b n |<ε+ε=2ε=ε 1 "n>N
Set "ε 1 >0, set ε=ε 1 /2. Then for any ε 1 >0 $N=maxN 1 N 2: " n>N Þ |a n +b n |<ε 1 Û lim(n ® ¥)(a n +b n)=0, то
is a n +b n - infinitely small.
Theorem The product of an infinitesimal is an infinitesimal.
a n ,b n is infinitely small Þ a n b n is infinitely small.
Proof:
Set "ε 1 >0, set ε=Öε 1 , since a n and b n are infinitely small for this ε>0, then there is N 1: " n>N Þ |a n |<ε
$N 2: " n>N 2 Þ |b n |<ε
Let's take N=max (N 1 ;N 2 ), then "n>N = |a n |<ε
|a n b n |=|a n ||b n |<ε 2 =ε 1
" ε 1 >0 $N:"n>N |a n b n |<ε 2 =ε 1
lim a n b n =0 Û a n b n is infinitesimal, which was to be proved.
Theorem The product of a bounded sequence and an infinitesimal sequence is an infinitesimal sequence
and n is a bounded sequence
a n is an infinitesimal sequence Þ a n a n is an infinitesimal sequence.
Proof: Since а n is bounded Û $С>0: "nн N z |a n |£C
We set "ε 1 >0; we set ε=ε 1 /C; since a n is infinitesimal, then ε>0 $N:"n>NÞ |a n |<εÞ |a n a n |=|a n ||a n | "ε 1 >0 $N: "n>N Þ |a n a n |=Cε=ε 1 Þ lim(n ® ¥) a n a n =0Û a n a n is an infinitesimal The sequence is called BBP(sequence) if Write . Obviously, the BBP is not limited. The converse statement is generally not true (example). If for large n members, then they write this means that as soon as . The meaning of the notation is defined similarly Infinitely large sequences a n =2 n ;
b n \u003d (-1) n 2 n ; c n \u003d -2 n Definition(infinitely large sequences) 1) lim(n ® ¥)a n =+¥ if "ε>0$N:"n>N Þ a n >ε where ε is arbitrarily small. 2) lim(n ® ¥)a n =-¥ if "ε>0 $N:"n>N Þ a n<-ε 3) lim(n ® ¥)a n =¥ Û "ε>0 $N:"n>N Þ |a n |>ε TICKET 7 Theorem “On convergence monoton. last" Any monotone sequence is convergent, i.e. has limits. Doc-in Let the last (xn) monotone ascend. and limited from above. X - all the set of numbers that takes the el-t of this last according to the conventions. Theorems are many limitations. Therefore, according to acc. Theorem, it has a finite exact top. the face supX xn®supX (we denote supX by x*). Because x* exact top. edge, then xn£x* " n. " e >0 vyp-sya $ xm (let m be n with a lid): xm>x*-e with " n>m => from the indicated 2 inequalities, we obtain the second inequality x*-e£xn£x*+e for n>m is equivalent to 1xn-x*1 TICKET 8 Exponent or number e R-rim num. last with a common term xn=(1+1/n)^n (to the power of n)(1) . It turns out that the sequence (1) increases monotonically, is bounded from above and is somewhat convergent, the limit of this post is called the exponential and is denoted by the symbol e "2.7128 ... Number e TICKET 9 The principle of nested segments Let a sequence of segments ,,…,,… be given on the number line Moreover, these segments satisfy the sl. cond.: 1) each successor is nested in the previous one, i.e. Ì, "n=1,2,…; 2) The lengths of segments ®0 with increasing n, i.e. lim(n®¥)(bn-an)=0. The sequence with the specified saints is called nested. Theorem Any sequence of nested segments contains a single m-ku with belonging to all segments of the sequence at the same time, with a common point of all segments to which they are contracted. Doc-in(an)-sequence of the left ends of the segments yavl. monotonically non-decreasing and bounded from above by the number b1. (bn)-sequence of right ends is monotonically non-increasing, therefore these sequences are yavl. converging, i.e. noun numbers с1=lim(n®¥)an and с2=lim(n®¥)bn => c1=c2 => c - their common value. Indeed, it has the limit lim(n®¥)(bn-an)= lim(n®¥)(bn)- lim(n®¥)(an) due to condition 2) o= lim(n®¥)(bn- an)=c2-c1=> c1=c2=c It is clear that m. c is common to all segments, since "n an£c£bn. Now we prove that it is one. Let's assume that $ is another c' to which all segments are contracted. If we take any non-intersecting segments c and c', then on the one hand the entire “tail” of the last (an), (bn) must be located in the vicinity of c'' (because an and bn converge to c and c' at the same time). Contradiction doc-et t-mu. TICKET 10 Bolzano-Weierstrass theorem
From any limit. last, you can choose a gathering. subseq. 1. Since the last is bounded, then $ m and M such that " m £ xn £ M, " n. D1= - the segment in which all the m-ki of the last lie. Let's split it in half. At least in one of the halves there will be an infinite number of so-to-last. D2 is the half where an infinite number of m-to-last lies. We split it in half. At least in one of the halves neg. D2 nah-Xia an infinite number of so-to the last. This half is D3. We divide the segment D3 ... and so on. we obtain a sequence of nested segments, the lengths of which tend to 0. According to the m-me about nested segments, $ unities. t-ka C, cat. owned all segments D1, some t-ku Dn1. In segment D2 I choose m-ku xn2, so that n2>n1. In segment D3 … etc. As a result, we eat the last xnkÎDk. TICKET 11 TICKET 12 fundamental In conclusion, consider the question of the criterion for the convergence of a numerical sequence. Let i.e.: We have received the following statement: If the sequence converges, then the condition Cauchy: A numerical sequence that satisfies the Cauchy condition is called fundamental. It can be proved that the converse is also true. Thus, we have a criterion (necessary and sufficient condition) for the convergence of a sequence. Cauchy criterion. For a sequence to have a limit, it is necessary and sufficient that it be fundamental. The second meaning of the Cauchy criterion. Sequence members and where n and m are any infinitely approaching at . TICKET 13 Unilateral limits. Definition 13.11. Number BUT is called the limit of the function y = f(x) at X striving for x 0 left (right) if such that | f(x)-A|<ε при x 0 - x< δ
(x - x 0< δ
). Designations: Theorem 13.1 (second definition of the limit). Function y=f(x) has at X, aspiring to X 0 , limit equal to BUT, if and only if both of its one-sided limits at this point exist and are equal BUT. Proof. 1) If , then and for x 0 - x< δ, и для x - x 0< δ
|f(x) - A|<ε, то есть 1) If , then there exists δ 1: | f(x) - A| < ε при x 0 – x< δ 1 и δ 2: |f(x) - A| < ε при x - x 0<
δ2. Choosing from the numbers δ 1 and δ 2 smaller and taking it for δ, we get that for | x-x0| < δ |f(x) - A| < ε, то есть . Теорема доказана. Comment. Since the equivalence of the requirements contained in the definition of the limit 13.7 and the condition for the existence and equality of one-sided limits is proved, this condition can be considered the second definition of the limit. Definition 4 (according to Heine) Number BUT is called the limit of the function if any BBP of argument values the sequence of corresponding function values converges to BUT. Definition 4 (according to Cauchy). Number BUT called if . It is proved that these definitions are equivalent. TICKET 14 and 15 Function limit properties at a point 1) If the limit exists in t-ke, then it is unique 2) If in the cycle x0 the limit of the function f(x) lim(x®x0)f(x)=A lim(x®x0)g(x)£B=> then in this t-ke $ the limit of the sum, difference, product and quotient. Separation of these 2 functions. a) lim(x®x0)(f(x)±g(x))=A±B b) lim(x®x0)(f(x)*g(x))=A*B c) lim(x®x0)(f(x):g(x))=A/B d) lim(x®x0)C=C e) lim(x®x0)C*f(x)=C*A Theorem 3. If a ( resp A ) then $ is the neighborhood in which the inequality >B(resp Setting in Theorem 3 B=0, we get: if ( resp), then $ , at all points, which will be >0 (resp<0),
those. the function retains the sign of its limit. Theorem 4(on passing to the limit in inequality). If in some neighborhood of the point (except perhaps for this point itself) the condition is satisfied and these functions have limits at the point, then . In the language and Let's introduce a function . It is clear that in a neighborhood of t. Then, by the function conservation theorem, we have the value of its limit, but Theorem 5.(on the limit of an intermediate function). (1) If TICKET 16 Definition 14.1. Function y=α(x) is called infinitesimal at x→x 0, if Properties of infinitesimals. 1. The sum of two infinitesimals is infinitesimal. Proof. If a α(x) and β(x) are infinitesimal for x→x 0, then there are δ 1 and δ 2 such that | α(x)|<ε/2 и |β(x)|<ε/2 для выбранного значения ε. Тогда |α(x)+β(x)|≤|α(x)|+|β(x)|<ε, то есть |(α(x)+β(x))-0|<ε. Следовательно, Comment. It follows that the sum of any finite number of infinitesimals is infinitesimal. 2. If α( X) is infinitely small at x→x 0, a f(x) is a function bounded in some neighborhood x 0, then α(x)f(x) is infinitely small at x→x 0. Proof. Choose a number M such that | f(x)| Corollary 1. The product of an infinitesimal by a finite number is an infinitesimal. Corollary 2. The product of two or more infinitesimals is an infinitesimal. Corollary 3. A linear combination of infinitesimals is infinitesimal. 3. (The third definition of the limit). If , then a necessary and sufficient condition for this is that the function f(x) can be represented as f(x)=A+α(x), where α(x) is infinitely small at x→x 0. Proof. 1)
Let Then | f(x)-A|<ε при x→x 0, i.e α(x)=f(x)-A is infinitely small at x→x 0 . Hence , f(x)=A+α(x). 2) Let f(x)=A+α(x). Then Comment. Thus, one more definition of the limit is obtained, which is equivalent to the two previous ones. Infinitely large features. Definition 15.1. The function f(x) is called infinitely large for x x 0 if For infinitely large one can introduce the same classification system as for infinitesimal ones, namely: 1. Infinitely large f(x) and g(x) are considered to be of the same order if 2. If , then f(x) is considered to be an infinitely large higher order than g(x). 3. An infinitely large f(x) is called a k-th order relative to an infinitely large g(x) if . Comment. Note that a x is an infinitely large (for a>1 and x ) higher order than x k for any k, and log a x is an infinitely lower order than any power of x k . Theorem 15.1. If α(x) is infinitely small for x→x 0 , then 1/α(x) is infinitely large for x→x 0 . Proof. Let us prove that for |x - x 0 |< δ. Для этого достаточно выбрать в качестве ε 1/M. Тогда при |x - x 0 | < δ |α(x)|<1/M, следовательно, |1/α(x)|>M. Hence, , that is, 1/α(x) is infinitely large as x→x 0 . TICKET 17 Theorem 14.7 (first remarkable limit). . Proof. Consider a circle of unit radius centered at the origin and assume that the angle AOB is x (radian). Let's compare the areas of the triangle AOB, the sector AOB and the triangle AOC, where the line OS is a tangent to the circle passing through the point (1; 0). It's obvious that . Using the corresponding geometric formulas for the areas of figures, we obtain from this that 
along with a natural number, one can substitute another natural number into the last inequality 
,then
and in some neighborhood of t. (except perhaps the t. itself) condition (2) is satisfied, then the function has a limit in t. and this limit is equal to BUT. by condition (1) $ for (here is the smallest neighborhood of the point ). But then, by virtue of condition (2), the value will also be located in - the vicinity of the point BUT, those. .
, i.e α(х)+β(х) is infinitesimal.
means | f(x)-A|<ε при |x-x0| < δ(ε). Cледовательно, .
, or sinx