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?  

There's a nice generalization to this: let $\,\{A_i\}_{i\in I}\,$ be a finite partition of the naturals $\,\Bbb N\,$ , with $\,|A_i|=\aleph_0\,\,\forall\,i\in I\,\,\,,\,|I|<\infty$ , and s.t. for a sequence $\,\{x_n\}\,$ there exists a number $\,\alpha\,$ s.t. we have
$$\lim_{m\to\infty}\{x_m\;\;|\;\;m\in I\}=\alpha\,\,,\,\,\forall\,\,i\in I$$
then
$$\lim_{n\to\infty}x_n=\alpha$$
The other way around is also true, of course, so the above can be put in iff form.
