Proof of Cauchy Sequence Convergence with Subsequence

In summary, the conversation discusses the convergence of a Cauchy sequence and its subsequence to a limit L. The attempt at a solution raises the question of whether a subsequence's convergence to L implies the convergence of the whole sequence to L. The expert responds by clarifying that this is only true if the sequence is known to converge, which is the case for a Cauchy sequence. The conversation then explores how assuming the sequence converges to a different limit leads to a contradiction.
  • #1
tarheelborn
123
0

Homework Statement



If {s_n} is a Cauchy sequence of real numbers which has a subsequence converging to L, prove that {s_n} itself converges to L.

Homework Equations




The Attempt at a Solution



I know that all Cauchy sequences are convergent, and I know that any subsequences of a convergent sequence are convergent to the same limit as the sequence, but I am not sure if I can turn the second part of the statement around to say that if a subsequence is convergent to L, then the sequence converges to the same limit. Any ideas? Thanks.
 
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  • #2
tarheelborn said:

Homework Statement



If {s_n} is a Cauchy sequence of real numbers which has a subsequence converging to L, prove that {s_n} itself converges to L.

Homework Equations




The Attempt at a Solution



I know that all Cauchy sequences are convergent, and I know that any subsequences of a convergent sequence are convergent to the same limit as the sequence, but I am not sure if I can turn the second part of the statement around to say that if a subsequence is convergent to L, then the sequence converges to the same limit. Any ideas? Thanks.
Actually, you can't just say "if a subsequence converges to L, then the sequence must converge to L". For example, the sequence [itex]\{a_n\}[/itex] with [itex]a_n= 1- 1/n[/itex] for n even and [itex]a_n= 1/n[/itex] for n odd has a subsequence ([itex]\{a_n\}[/itex] for a even) that converges to 1 but the sequence itself does not converges.

What you can say is that if the sequence converges, then because, as you say, all subsequences must converge to that limit, yes, the sequence converges to whatever limit any subsequence converges to. The difference is that you must know the sequence does converge. Which you know here because it is a Cauchy sequence.
 
  • #3
Well, suppose that the Cauchy sequence converges to a number M different than L. Show that this leads to a contradiction by showing that there exists an epsilon such that some sequence value is always further away from M than epsilon.
 
  • #4
Thank you!
 

What is a Cauchy sequence?

A Cauchy sequence is a sequence of real numbers that satisfies the Cauchy criterion, which states that for any small positive number, there exists a point in the sequence after which all the terms are within that small distance of each other.

What is the importance of proving a sequence is Cauchy?

Proving that a sequence is Cauchy is important because it guarantees that the sequence converges to a limit. This is useful in many areas of mathematics, particularly in the study of real numbers and their properties.

What is the proof for a Cauchy sequence?

The proof for a Cauchy sequence involves showing that for any small positive number, there exists a point in the sequence after which all the terms are within that small distance of each other. This can be done by using the definition of a Cauchy sequence and manipulating the terms of the sequence.

What are some common techniques used in proving Cauchy sequences?

Some common techniques used in proving Cauchy sequences include the triangle inequality, the Archimedean property, and the completeness property of the real numbers. These properties help to manipulate the terms of the sequence and show that they are within a small distance of each other.

What are some common mistakes in a Cauchy sequence proof?

Some common mistakes in a Cauchy sequence proof include not using the correct definition of a Cauchy sequence, not considering all possible values for the small positive number, and making incorrect assumptions about the terms of the sequence. It is important to carefully follow the steps of the proof and double check all assumptions and calculations.

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