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I very much fear the answer here is going to be quite simple / obvious (or the other end of the spectrum and very open), but I can't quite wrap my head around things, so here goes.

Say we have a sequence $(a_n)$, and it has a subsequence $(a_{n_i})$ which converges to some limit, say $L$.

Are there any necessary conditions on $(a_n)$, or perhaps $(a_{n_i})$, which would allow us the conclude that $(a_n)$ also converges to $L$?

I guess it should go without saying that $(a_n)$ must be bounded. $(a_n)$ being increasing, or decreasing, seems to be sufficient, but certainly not necessary. I'm tempted to think that $(a_n)$ being Cauchy might be necessary, but I must confess that my knowledge of sequences and limits is somewhat rusty, so I'm worried I might be overlooking something there.

So is there anything else which might help here? Even a source to help explain these sorts of scenarios a bit more would be of great use to me!

Mystery_Jay
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You really need any condition that guarantees that $(a_n$) has a limit. Once it has a limit, you know it is equal to any subsequence limits. So yes, monotone and bounded is one such condition.

Alan
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  • "Once it has a limit, you know it is equal to any subsequence limits." This must be something that is pretty common knowledge that I've either forgotten about or didn't realise - how exactly do we know this? Is this a consequence of the Bolzano-Weierstass Theorem? – Mystery_Jay May 10 '21 at 15:45
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    @Mystery_Jay https://math.stackexchange.com/questions/213285/prove-if-a-sequence-converges-then-every-subsequence-converges-to-the-same-lim – Alan May 10 '21 at 15:56