I am trying to prove the following proposition: "If $x_n$ is a convergent sequence, then every subsequence of $x_n$ is convergent and converges to the same limit as $x_n$."
I am not looking for an answer - I would not like a direct answer - but rather some guidance on how to prove this.
Firstly, I think I need to show that every subsequence of $x_n$ is convergent. So let $x_{n_r}$ be a subsequence. By definition, the $n_r's$ are strictly increasing, so can I deduce from here that the subsequence $x_{n_r}$ is strictly increasing as well?
I know as well that as $x_n$ is convergent, it is bounded, viz $|x_n|\leq M$ where $M > 0$. So as the terms in a subsequence are contained in the set of all the $x_n's$, this means that every subsequence of $x_n$ is bounded as well?
If I can deduce that the $x_{n_r}'s$ are bounded and monotone, then I know that every subsequence of a convergent sequence is convergent.
Now the hard part of showing that every subsequence converges to the same limit, of which I have no idea; I could begin though to assume the negation that there exists a subsequence such that it converges to a different limit, say $M$ while the $x_n's$ converge to $L$ instead.