Sufficiency to prove the convergence of a sequence using even and odd terms Given a sequence $a_{n}$, if I know that the sequence of even terms converges to the same limit as the subsequence of odd terms:  
$$\lim_{n\rightarrow\infty} a_{2n}=\lim_{n\to\infty} a_{2n-1}=L$$
Is this sufficient to prove that the $\lim_{n\to\infty}a_{n}=L$?  
If so, how can I make this more rigorous?  Is there a theorem I can state that covers this case?  

If you are familiar with subsequences, you can easily prove as follows. Let $a_{n_k}$ be the subsequence which converges to $\limsup a_n$. it is obviously convergent and contain infinitely many odds or infinitely many evens, or both. Hence, $\limsup a_n = L$. The same holds for $\liminf a_n$, hence the limit of the whole sequence exists and equals $L$.
