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.
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Note that it suffices to show that, $$ \lim a_{2n} = \lim a_{2n + 1}.$$ As 
$ a_{6n} $ is a subsequence of $a_{2n}$ and $a_{3n}$. We have $ \lim a_{2n} = \lim a_{3n}$ And as $a_{3n} = a_{2n_m + 1} $. We have $$\lim a_{2n + 1} = \lim a_{3n} = \lim a_{2n}. $$ 
