Subsequences and convergence. Do all subsequences have to converge to the same limit for the sequence to be convergent? I was asked this question:
Prove that $a_n$ converges if and only if:
$a_{2n},a_{2n+1},a_{3n}$ all converge
I thought this was an easy generic question until I read the hint which said:
Note: It is not required that the three sub-sequences have the same limit. This needs to be shown
This is what is confusing me because I have found two sources stating something different:
Proposition 4.2. A sequence an converges to L ∈ R if and only if every subsequence converges to L.
and
Let $a_n$ be a real sequence. If the subsequence $a_{2n}$ converges to a real number L and the subsequence $a_{2n+1}$ converges to the same number L, then $a_n$ converges to L as well.
So my question is: for a sequence $a_n$ to converge does it's subsequences have to converge to the same limit? (I suspect not) and if the answer is no can you help me prove why?
Thanks in advance 

I think the word "required" in the hint is confusing. Focus instead on the last sentence:

Note: It is not required that the three sub-sequences have the same limit. This needs to be shown

In other words, you need to show that as a result of the given hypotheses, the three sub-sequences do have the same limit. You need to do that precisely because it is a necessary condition for $a_n$ to converge.
