# Convergence of two subsequences

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?

• Well, that seems almost equivalent to what you are asked to prove. If you already have that theorem as something you can quote then, sure. But otherwise, I'd say you still need to add more detail.
– lulu
Oct 2 '19 at 20:16

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.