real analysis - Necessary and sufficient condition for the existence of $lim_{ntoinfty} a_n$

Let {$a_n$} be a sequence of real numbers. Then what should be a necessary and sufficient condition for the existence of $\lim_{n\to\infty} a_n$?

P.S.- The condition should be in terms of existence of other limits, for example, it exists if $\lim_{n\to\infty} a_{2n}$ and $\lim_{n\to\infty} a_{3n}$ both exist, say...