- #1

Hall

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*standard*proof:

Given ##\epsilon \gt 0## there exists an ##N## such that

$$

k \gt N \implies |s_k - s| \lt \varepsilon$$

Consider any subsequence ##(s_{n_{k}})##. Since ##n_k \geq k##, therefore ## k \gt N \implies n_k \gt N##. Hence,

$$

k \gt N \implies |s_{n_{k}} - s| \lt \varepsilon$$

So, we can write ## \lim s_{n_{k}} = s##.

But it seems to me that this proof is more of a consequence of the language than logic, I know the idea of this proof is logical: if all the terms after a certain point gets nearer and nearer to a point, then no matter how many finite terms we delete, the higher terms will still be near to that same point (because subsequence also contains infinite terms, and eventually the higher terms will get to ##s## only); but this rigorous proof seems merely to be language game.

If we write the subsequences as ##t_n## (and not as ##s_{n_k}##), then how we would proceed to show ##\lim t_n = s##?