If Every sub sequence of a sequence $\left\{a_n\right\}$ converges to the same limit, then the sequence $\left\{a_n\right\}$ also converges to the same limit.
This might be in lot of threads but i want to know whether my proof has any flaws.
Given that the sub sequence $\left\{a_{{n_k}}\right\}$ is convergent to a limit say $L$, we have, for a given $\epsilon >0$, $\exists K \in \mathbb{N}$, such that $$|a_{{n_k}}-L|<\epsilon$$ whenever $n_k \geq K$
Now let us assume that the sequence $\left\{a_n\right\}$ does not converge. So for a given $\epsilon >0$, we can always find $n_0 \in \mathbb{N}$ such that $$|a_n-L|>\epsilon$$ whenever $n \geq n_0$
Now letting $M=max(K,n_0)$ we have $\forall n \geq M$ $$\epsilon <|a_n-L|<\epsilon$$ which is false. Hence a contradiction. Thus the sequence $\left\{a_n\right\}$ should converge.
Is this proof fine?