Convergence of a Sequence by its Subsequences Question Given the sequence $\{a_n\}_{n=0}^\infty$ and the subsequences: $a_{3n}, a_{2n+1}, a_{2n}$ which converge. Prove that $a_n$ is a convergent sequence.
Thanks you very much.

Hint: first show that the three subsequences have the same limit. (The subsequence $(a_{3n})$ has a further subsequence that is a subsequence of $(a_{2n})$, for instance.)
Then note that given $n>1$, $a_n$ is a term of one of the   subsequences $(a_{2n})$, $(a_{2n+1})$. (So, given $\epsilon>0$, choose $N$ so that for any $n\ge N$, each of $a_{2n}$ and $a_{2n+1}$  is within $\epsilon$ of the common limit. Then... .)
