Given subsequences converge, prove that the sequence converges.

Hint: it is sufficient to show that the convergent sequences $\{a_{2k}\}$ and $\{a_{2k+1}\}$ have the same limit. To see that they do have the same limit, note that $\{a_{2k}\}$ and $\{a_{2k+1}\}$ each have a subsequence that is a subsequence of the convergent sequence $\{a_{3k}\}$.


If all of the even subsequences converge and all of the odd subsequences converge, then the original sequence will converge iff both of the above subsequences converge to the same value.

To show both the even and odd subsequences converge to the same value, you need a third subsequence that converges and covers more than a finite amount of even and odd values. That third subsequence is given here as $\{a_{3k}\}$.

More generally, if two subsequences converge and every term in the original sequence belongs to one of the two subsequences, then convergence of the original sequence requires a third sequence that takes terms from both sequences infinitely many times.

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Real Analysis

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