I have looked through previous posts but have been struggling with this problem.
The sequence is {$a_n$} and its subsequences {$a_{2k}$}, {$a_{2k+1}$}, {$a_{3k}$} converge. I have to prove that {$a_n$} converges.
I know that a sequence converges if all of its subsequences converge. I'm suspecting that I have to prove that every subsequence belongs into these 3 subsequences. Thank you for your time and help.
Best Answer
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 $\{a_{3k}\}$.