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Scousergirl

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

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- Thread starter Scousergirl
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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.

- #1

Scousergirl

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

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morphism

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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.

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.

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.

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.

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.

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