Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge? [closed]

Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge?
$a_{2n}$ is a subsequence of $a_n$ and $a_{3n},a_{2n}$ are subsequences of $a_{6n}$ so they all have the same limit
But what about $a_{2n-1}$ can I say it is a subsequence of $a_{2n}$ and therefore they all have the same limit?

Solution 1:

Yes. Indeed one direction is easy, if $a_n$ converges then all its subsequences converge to the same limit.

For the other direction say $a_{2n}\to p$, $a_{2n-1}\to q$, and $a_{3n}\to r$. Since elements of $a_{3n}$ occur infinitely often in both sequences $a_{2n}$ and $a_{2n-1}$, it follows that $r=p$ and $r=q$, so also $p=q$. Since the two sequences $a_{2n}$ and $a_{2n-1}$ exhaust $a_n$ it follows that $a_n\to p$ too. (Indeed take any $\varepsilon >0$ then there are $i$ and $j$ such that if $2n>i$ then $|a_{2n}-p|<\varepsilon$, and if $2n-1>j$ then $|a_{2n-1}-p|<\varepsilon$. If follows that if $n>k:=\max\{i,j\}$ (even or odd) then $|a_{n}-p|<\varepsilon$).

I have answered the question as stated in the title (which seems the only place where it is stated):

Does $a_n$ converge if and only if $a_{2n},a_{3n},a_{2n−1}$ converge?