(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that [tex](x_n)[/tex] is a sequence in a compact metric space with the property that every convergent subsequence has the same limit [tex]x[/tex]. Prove that [tex]x_n \to x[/tex] as [tex]n\to \infty[/tex]

2. Relevant equations

Not sure, most of the relevant issues pertain to the definitions of the space. In this case I believe the following is relevant:

In a compact metric space every sequencemusthave a convergent subsequence, defining it as sequentially compact.

I'll add more.

3. The attempt at a solution

My basic hang up is this: Does every subsequence have to be convergent? If so...you can.

Suppose not:

Assume there exists a subsequence [tex]s_n[/tex] in [tex](x_n)[/tex] s.t. [tex]s_m =\displaystyle\sum\limits_{k=1}^{m} x_k \to y \neq x[/tex]

Then there would exist an [tex]m>a>0[/tex]

Under the assumption that [tex]s_a = \displaystyle\sum\limits_{k=a}^{\infty} x_k \to x[/tex]

Where [tex]s_m = \displaystyle\sum\limits_{k=1}^{m+a} x_k[/tex] Would not be convergent.

Then the sequence [tex](x_n)[/tex] is not Cauchy, which implies it's not a Complete Metric Space.

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# Convergent Subsequences in Compact Metric Space

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