Bounded sequence implies convergent subsequence

Thread starter Scousergirl
Start date Sep 19, 2007
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Bounded Convergent Sequence Subsequence

In summary, a bounded sequence is a sequence of numbers that falls within a certain interval or is limited to a certain value. This implies the existence of a convergent subsequence, which is a subset of the original sequence that approaches a specific limit or value. A convergent subsequence is different from a convergent sequence in that not all terms in the original sequence need to approach the limit. A bounded sequence can have multiple convergent subsequences, and this concept is important in mathematics as it helps prove convergence and understand the behavior of a sequence.

Sep 19, 2007

Scousergirl

How can you deduce that nad bounded sequence in R has a convergent subsequence?

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Sep 19, 2007

morphism

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Homework Helper

This is the Bolzano–Weierstrass theorem. Any decent calculus or analysis textbook should have a proof of it. Or search online. Or use this hint: Prove that a bounded sequence of real numbers has a monotone subsequence.

FAQ: Bounded sequence implies convergent subsequence

1. What is a bounded sequence?

A bounded sequence is a sequence of numbers that is limited in range. This means that all the values in the sequence fall within a certain interval, or they are not larger than a certain number.

2. What does it mean for a bounded sequence to imply a convergent subsequence?

This means that within the bounded sequence, there is a subsequence (a sequence of numbers within the original sequence) that converges to a specific limit or value. In other words, as the terms in the subsequence approach infinity, they get closer and closer to a certain number.

3. How is a convergent subsequence different from a convergent sequence?

A convergent sequence is a sequence where all the terms approach a single limit or value. A convergent subsequence, on the other hand, is a subset of a larger sequence that also approaches a limit or value, but not all the terms in the original sequence need to do so.

4. Can a bounded sequence have more than one convergent subsequence?

Yes, a bounded sequence can have multiple convergent subsequences. As long as there is a subset of the original sequence that approaches a limit or value, it can be considered a convergent subsequence.

5. Why is the concept of a bounded sequence implying a convergent subsequence important in mathematics?

This concept is important because it allows us to prove the convergence of a sequence without knowing its exact limit or value. It also helps us understand the behavior of a sequence and make predictions about its convergence based on its boundedness.