# Bounded sequence, convergent subsequence

• kathrynag
In summary, a bounded sequence is a sequence of numbers with a finite upper and lower limit, while a convergent subsequence is a subsequence of a bounded sequence that approaches a specific limit. To determine if a sequence is bounded, a number M must exist where all terms in the sequence are less than or equal to M. To find a convergent subsequence in a bounded sequence, the Bolzano-Weierstrass theorem can be used. These concepts are important in understanding the behavior of infinite sequences and have many applications in mathematics.
kathrynag

## Homework Statement

Asssume (an) is a bounded sequence with the property that every convergent subsequence of (an) converges to the same limit a. Show that (an) must converge to a.

## The Attempt at a Solution

If the subsequence converges to a we have , we have:
$$\left|ank-a\right|$$=a.

Choose $$n_k$$ large enough such that $$n_k \geq k \geq n_o$$. You should be to use this to show that your sequence converges.

1.

## What is a bounded sequence?

A bounded sequence is a sequence of numbers that has a finite upper and lower limit. This means that the values in the sequence do not exceed a certain range.

2.

## What is a convergent subsequence?

A convergent subsequence is a subsequence of a bounded sequence that approaches a specific limit as the number of terms in the subsequence increases.

3.

## How do you determine if a sequence is bounded?

A sequence is bounded if there exists a number M such that all the terms in the sequence are less than or equal to M. This means that the values in the sequence do not exceed a certain range.

4.

## How do you find a convergent subsequence in a bounded sequence?

To find a convergent subsequence in a bounded sequence, you can use the Bolzano-Weierstrass theorem. This theorem states that every bounded sequence has a convergent subsequence.

5.

## What is the importance of bounded sequences and convergent subsequences in mathematics?

Bounded sequences and convergent subsequences are important in mathematics because they help us understand the behavior of infinite sequences. They also have many applications in fields such as calculus, analysis, and numerical analysis.

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