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

2. Relevant equations

N/A

3. The attempt at a solution

Actually I read over this question for about 10 times already. But, I am not sure how to start. I know that I have to construct a subsequence one term at a time and show that it converges to L. But all the different subscripts and indices are just driving me crazy. So our target subsequence (the one that we need to construct at the end) is denoted as (x_{nk}).

We are given that "For each k≥1, there is a subsequence of (x_{n}) converging to L_{k}...". How should I denote this subsequence, then? I was thinking of (x_{ni}), but I think this would be the same as our target subsequence (x_{nk}) becuase the i and k are just dummy variables. Also, it is "for EACH k≥1...", so this subsequence also has some dependence on k, so I think k should appear as part of the subscript as well? How should I label this subsequence properly?

So I am stuck even at the level of translating the question into mathematical symbols and summarizing what is given. Can someone please show me how?

I hope someone can help me out.

Thank you!

[note: also under discussion in Math Links forum]

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# Homework Help: Show there exists a SUBSEQUENCE converging to L

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