[note: also under discussion in Math Links forum]
** There is a subs. converging to . Choose now .
** There is a subs. converging to . Choose now in such a way that (we can do this because we've an infinite subsequence here and thus there exists !)
** There is a subs. converging to . Choose now in such a way that (we can do this because we've an infinite subsequence here and thus there exists !)
Continue inductively in the above way, and in the m-th step:
** There is a subs. converging to . Choose now in such a way that (we can do this because we've an infinite subsequence here and thus there exists !)
Take it from here...
Tonio
Thanks for your great help! But I still have some concerns about extracting the information given in the problem and the notations for different subsequences.
As stated in the problem, we are using the notation (x_n_k) for the target subsequence (the one that we need to construct at the end), so I'm going to stick with that.
"For each k≥1, there is a subsequence of (x_n) converging to L_k...".
So if you're using (x_n_k (1) ), (x_n_k (2) ), ..., (x_n_k (m) ),..., wouldn't this notation mean that it is a (further) subsequence of our target subsequence (x_n_k)???
Also the question says "For each k≥1...", so I think k should occur in the bracket (x_n_k (k) ), right? But then, we'll have two k's in it. Is that OK?
So now I am confused with all those subscripts, indices and the relation between them. k seems to have a special meaning in this question that is tied to the subscript of our target subsequence. So we are going to fix k=1,k=2,k=3,... one at a time to construct our target subsequence.
What is the usual (and correct) way to denote the DIFFERENT subsequences of (x_n)?
Thanks for clarifying this! It has been a nightmare for me...