Let the variable x n takes an infinite sequence of values

x 1 , x 2 , ..., x n , ..., (1)

and the law of change of variable is known x n, i.e. for every natural number n you can specify the appropriate value x n. Therefore, it is assumed that the variable x n is a function of n:

x n = f(n)

Let us define one of the most important concepts of mathematical analysis - the limit of a sequence, or, what is the same, the limit of a variable x n, running through the sequence x 1 , x 2 , ..., x n , ... . .

Definition. Constant number a called limit of the sequence x 1 , x 2 , ..., x n , ... . or the limit of a variable x n, if for an arbitrarily small positive number e there is such a natural number N(i.e. number N) that all values ​​of the variable x n, beginning with x N, differ from a in absolute value less than by e. This definition is briefly written as follows:

| x n -a |< (2)

in front of everyone n  N, or, what is the same,

Determination of the Cauchy limit. A number A is called the limit of a function f (x) at a point a if this function is defined in some neighborhood of the point a, with the possible exception of the point a itself, and for every ε > 0 there exists δ > 0 such that for all x satisfying condition |x – a|< δ, x ≠ a, выполняется неравенство |f (x) – A| < ε.

Determination of the Heine limit. A number A is called the limit of a function f (x) at a point a if this function is defined in some neighborhood of the point a, with the possible exception of the point a itself, and for any sequence such that converging to the number a, the corresponding sequence of function values ​​converges to the number A.

If a function f (x) has a limit at point a, then this limit is unique.

The number A 1 is called the limit of the function f (x) on the left at point a if for every ε > 0 there exists δ >

The number A 2 is called the limit of the function f (x) on the right at point a if for each ε > 0 there exists δ > 0 such that the inequality holds for all

The limit on the left is denoted by the limit on the right - These limits characterize the behavior of the function to the left and right of point a. These are often called one-way limits. In the designation of one-sided limits for x → 0, the first zero is usually omitted: and . So, for the function

If for every ε > 0 there exists a δ-neighborhood of a point such that for all x satisfying the condition |x – a|< δ, x ≠ a, выполняется неравенство |f (x)| >ε, then they say that the function f (x) has an infinite limit at point a:

Thus, the function has an infinite limit at the point x = 0. Limits equal to +∞ and –∞ are often distinguished. So,

If for every ε > 0 there is a δ > 0 such that for every x > δ the inequality |f (x) – A|< ε, то говорят, что предел функции f (x) при x, стремящемся к плюс бесконечности, равен A:

Existence theorem for an exact supremum

Definition:АR mR, m is the upper (lower) face of А, if аА аm (аm).

Definition: A set A is bounded from above (from below), if there exists an m such that aA, am (am) holds.

Definition: SupA=m, if 1) m is the supremum of A

2) m’: m’ m’ is not the supremum of A

InfA = n, if 1) n is the infimum of A

2) n’: n’>n => n’ is not the infimum of A

Definition: SupA=m is a number such that: 1)  aA am

2) >0 a  A, such that a  a-

InfA = n is a number such that: 1) 1)  aA an

2) >0 a  A, such that a E a+

Theorem: Any non-empty set AR bounded from above has an exact supremum, and a unique one.

Proof:

Let's construct the number m on the number line and prove that this is the supremum of A.

[m]=max([a]:aA) [[m],[m]+1]A=>[m]+1 - upper bound of A

Segment [[m],[m]+1] - divided into 10 parts

m 1 =max:aA)]

m 2 =max,m 1:aA)]

m k =max,m 1 ...m K-1:aA)]

[[m],m 1 ...m K , [m],m 1 ...m K + 1 /10 K ]A=>[m],m 1 ...m K + 1/ 10 K - top edge A

Let us prove that m=[m],m 1 ...m K is the supremum and that it is unique:

k: )