A proof of Weierstrass's theorem on the limit of a monotone sequence is given. The cases of bounded and unbounded sequences are considered. An example is considered in which it is necessary, using Weierstrass's theorem, to prove the convergence of a sequence and find its limit.
ContentSee also: Limits of monotonic functions
Any monotone bounded sequence (xn) has a finite limit equal to the exact upper limit, sup(xn) for a non-decreasing and exact lower bound, inf(xn) for a non-increasing sequence.
Any monotonic unbounded sequence has an infinite limit equal to plus infinity for a non-decreasing sequence and minus infinity for a non-increasing sequence.
Proof
1) non-decreasing bounded sequence.
(1.1)
.
Since the sequence is bounded, it has a finite upper bound
.
It means that:
- for all n,
(1.2) ; - for any positive number, there is a number depending on ε, so that
(1.3) .
.
Here we also used (1.3). Combining with (1.2), we find:
at .
Since then
,
or
at .
The first part of the theorem has been proven.
2)
Let now the sequence be non-increasing bounded sequence:
(2.1)
for all n.
Since the sequence is bounded, it has a finite lower bound
.
This means the following:
- for all n the following inequalities hold:
(2.2) ; - for any positive number, there is a number, depending on ε, for which
(2.3) .
.
Here we also used (2.3). Taking into account (2.2), we find:
at .
Since then
,
or
at .
This means that the number is the limit of the sequence.
The second part of the theorem is proven.
Now consider unbounded sequences.
3)
Let the sequence be unlimited non-decreasing sequence.
Since the sequence is non-decreasing, the following inequalities hold for all n:
(3.1)
.
Since the sequence is non-decreasing and unbounded, it is unbounded on the right side. Then for any number M there is a number, depending on M, for which
(3.2)
.
Since the sequence is non-decreasing, then when we have:
.
Here we also used (3.2).
.
This means that the limit of the sequence is plus infinity:
.
The third part of the theorem is proven.
4) Finally, consider the case when unbounded non-increasing sequence.
Similar to the previous one, since the sequence is non-increasing, then
(4.1)
for all n.
Since the sequence is non-increasing and unbounded, it is unbounded on the left side. Then for any number M there is a number, depending on M, for which
(4.2)
.
Since the sequence is non-increasing, then when we have:
.
So, for any number M there is a natural number depending on M, so that for all numbers the following inequalities hold:
.
This means that the limit of the sequence is equal to minus infinity:
.
The theorem has been proven.
Example of problem solution
All examples Using Weierstrass's theorem, prove the convergence of the sequence:
,
,
. . . ,
,
. . .
Then find its limit.
Let's represent the sequence in the form of recurrent formulas:
,
.
Let us prove that the given sequence is bounded above by the value
(P1) .
The proof is carried out using the method of mathematical induction.
.
Let . Then
.
Inequality (A1) is proven.
Let us prove that the sequence increases monotonically.
;
(P2) .
Since , then the denominator of the fraction and the first factor in the numerator are positive. Due to the limitation of the terms of the sequence by inequality (A1), the second factor is also positive. That's why
.
That is, the sequence is strictly increasing.
Since the sequence is increasing and bounded above, it is a bounded sequence. Therefore, according to Weierstrass's theorem, it has a limit.
Let's find this limit. Let's denote it by a:
.
Let's use the fact that
.
Let's apply this to (A2), using the arithmetic properties of limits of convergent sequences:
.
The condition is satisfied by the root.
Definition: if everyone n є N, compliant x n є N, then they say that
form numerical subsequence.
- members sequences
- general member sequences
The introduced definition implies that any number sequence must be infinite, but does not mean that all members must be distinct numbers.
The number sequence is considered given, if a law is specified by which any member of the sequence can be found.
Members or sequence elements (1) numbered by all natural numbers in ascending order. For n+1 > n-1, the term follows (precedes) the term, regardless of whether the number itself is greater than, less than, or even equal to the number.
Definition: A variable x that takes on some sequence (1) values, we - following Meray (Ch. Meray) - will call option.
In a school mathematics course you can find variables of exactly this type, such as options.
For example, a sequence like
(arithmetic) or type
(geometric progression)
The variable term of one or another progression is option.
In connection with determining the length of a circle, we usually consider the perimeter of a regular polygon inscribed in the circle, obtained from a hexagon by successively doubling the number of sides. Thus, this option takes the following sequence of values:
Let us also mention the decimal approximation (by disadvantage) to, with increasing accuracy. It takes a sequence of values:
and also presents the option.
The variable x, running through the sequence (1), is often denoted by, identifying it with the variable (“common”) member of this sequence.
Sometimes the option x n is specified by directly indicating the expression for x n ; so, in the case of an arithmetic or geometric progression, we have, respectively, x n =a+(n-1) d or x n =aq n-1. Using this expression, you can immediately calculate any variant value based on its given number, without calculating previous values.
For the perimeter of a regular inscribed polygon, such a general expression is possible only if we introduce the number p; in general, the perimeter p m of a regular inscribed m-gon is given by the formula
Definition 1: A number sequence (x n) is said to be bounded above (below) if such a number exists M (T), that for any element of this sequence there is an inequality, and the number M (m) is called top (lower) edge.
Definition 2: A number sequence (x n) is called bounded if it is bounded both above and below, i.e. there exist M, m, such that for any
Let us denote A = max (|M|, |m|), then it is obvious that the numerical sequence will be limited if for any the equality |x n |? And the last inequality is the condition for the limitedness of the numerical sequence.
Definition 3: a number sequence is called endlessly big sequence, if for any A>0, you can specify a number N such that for all n>N ||>A holds.
Definition 4: the number sequence (b n) is called endlessly small sequence, if for any given e > 0, you can specify a number N(e) such that for any n > N(e) the inequality | b n |< е.
Definition 5: the number sequence (x n) is called convergent, if there is a number a such that the sequence (x n - a) is an infinitesimal sequence. At the same time, a - limit original numerical sequences.
From this definition it follows that all infinitesimal sequences are convergent and the limit of these sequences = 0.
Due to the fact that the concept of a convergent sequence is linked to the concept of an infinitesimal sequence, the definition of a convergent sequence can be given in another form:
Definition 6: the number sequence (x n) is called convergent to a number a, if for any arbitrarily small there is such that for all n > N the inequality
a is the limit of the sequence
Because is equivalent, and this means that it belongs to the interval x n є (a - e; a+ e) or, which is the same, belongs to e - the neighborhood of point a. Then we can give another definition of a convergent number sequence.
Definition 7: the number sequence (x n) is called convergent, if there is a point a such that in any sufficiently small e-neighborhood of this point there are any elements of this sequence, starting from some number N.
Note: according to definitions (5) and (6), if a is the limit of the sequence (x n), then x n - a is an element of an infinitesimal sequence, i.e. x n - a = b n, where b n is an element of an infinitesimal sequence. Consequently, x n = a + b n, and then we have the right to assert that if a numerical sequence (x n) converges, then it can always be represented as the sum of its limit and an element of an infinitesimal sequence.
The converse statement is also true: if any element of the sequence (x n) can be represented as the sum of a constant number and an element of an infinitesimal sequence, then this constant is limit given sequences.
Definition 8. Sequence Not increases (not decreases), if for.
Definition 9. Sequence increases (decreasing), if for.
Definition 10. A strictly increasing or strictly decreasing sequence is called monotonous sequence.