# Convergence of a sequence clarification

Let’s say I have a sequence of real numbers, and I prove that the subsequence of even terms and the subsequence of odd terms both converge but not necessary to the same limit. Does that imply that the sequence converges? I am able to prove that if the above subsequences both converge to the same limit then the sequence also converges to that limit , but not quite sure about the above question. Thanks :)

• No in general. Consider the sequence $a_n=(-1)^n$. – Yadati Kiran Apr 19 at 18:16

To conclude that the original sequence converges, we need that the even and odd subsequences converge to the same limit. Indeed, consider the sequence $$a_n:=(-1)^n$$. Then the even subsequence $$a_{2n}=(-1)^{2n}=1$$ converges to $$1$$ and the odd subsequence $$a_{2n+1}=(-1)^{2n+1}=-1$$ converges to $$-1$$. However, the sequence $$a_n$$ does not converge to a limit.