Say $a_n$ is a sequence and there exits two converging subsequences $a_{n_k}$ and $b_{n_k}$ such that $\lim_{n ->\infty}a_{n_k} \neq \lim_{n ->}b_{n_k}$. I need to prove that $a_n$ does not converge.
In this question, I think proof by contradiction is a good approach. I suppose $a_n$ is a convergent sequence. Thus, if a sequence is convergent, then all of its subsequences are convergent and have the same limit, meaning that $\lim_{n ->\infty}a_{n_k} = \lim_{n ->}b_{n_k}$. There is a contradiction here.
Is my proof correct?
Best Answer
If you are given that "if a sequence is convergent, then all of its subsequences are convergent and have the same limit" your proof by contradiction is fine.
Otherwise you can start from the definition of limit for the sequence and for the two subsequences. Then choose $\epsilon_a$ and $\epsilon_n$ in such way that we obtain a contradiction for the limit of the sequence.