Show there exists a SUBSEQUENCE converging to L

[kingwinner]

Homework Statement

Homework Equations

N/A

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) because 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]

 

[ystael]

Notation is yours to define as you see fit. In particular, you're not required to conform to the notation in the hint if it confuses you.

Start by giving names to the indices of the subsequences of $$(x_n)$$ converging to each $$L_k$$; you might call these indices $$n_{k,j}$$, so that for each $$k$$, $$(n_{k,j})_{j=1}^\infty$$ is a strictly increasing sequence of natural numbers such that $$(x_{n_{k,j}})_{j=1}^\infty$$ converges to $$L_k$$. Your task is now to use these $$n_{k,j}$$ to choose a strictly increasing sequence $$(m_j)_{j=1}^\infty$$ such that $$(x_{m_j})_{j=1}^\infty$$ converges to $$L$$.

 

[kingwinner]

Notation is yours to define as you see fit. In particular, you're not required to conform to the notation in the hint if it confuses you.

Start by giving names to the indices of the subsequences of $$(x_n)$$ converging to each $$L_k$$; you might call these indices $$n_{k,j}$$, so that for each $$k$$, $$(n_{k,j})_{j=1}^\infty$$ is a strictly increasing sequence of natural numbers such that $$(x_{n_{k,j}})_{j=1}^\infty$$ converges to $$L_k$$. Your task is now to use these $$n_{k,j}$$ to choose a strictly increasing sequence $$(m_j)_{j=1}^\infty$$ such that $$(x_{m_j})_{j=1}^\infty$$ converges to $$L$$.

Sorry, I'm confused...

1) But if you write (x_nkj), it would mean that it is a (further) subsequence of (x_nk), right? But (x_nk) is our target subsequence, and I don't think there is such a connection...

2) Why are you using "m" in x_mj (target subsequence)?
Also, shouldn't there be some dependecy on k? (as the hint suggested?)

Maybe someone can clarify this, please? What is the usual (and correct) way to denote the DIFFERENT subsequences of (x_n)?

Thanks a lot!