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(i) Assuming that $(x_n)_n$ converges, investigate whether $(a_n)_n$ converges.

(ii)Assuming that $(a_n)_n$ converges, investigate whether $(x_n)_n$ converges.

For (ii):

If we choose $x_n=(-1)^n$, then $(a_n)_n$ converges as $(x_n)_n$ doesn't converge. And if we choose $x_n=1$, then $(a_n)_n$ converges as $(x_n)_n$ converges. So for part (ii), $(x_n)_n$ can both converge or not converge assuming that $(a_n)_n$ converges.

For (i):

I believe $(a_n)_n$ should be always converging, but I couldn't prove it, any help?

Thanks for your effort and time in advance.

Edit: I think the following question says that $(x_n)_n$ always converges when $(a_n)_n$ converges. Is my reasoning in part (ii) wrong? Prove $x_n$ converges if $x_n$ is a real sequence and $s_n=\frac{x_0+x_1+\cdots+x_n}{n+1}$ converges

user666150
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    The first part is answered (for example) here: https://math.stackexchange.com/questions/155839/on-cesàro-convergence-if-x-n-to-x-then-z-n-fracx-1-dots-x-nn – Martin R Oct 30 '20 at 21:31
  • That link answers the first part but can you check my edit before closing as duplicate – user666150 Oct 30 '20 at 21:35
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    Your examples for (ii) are fine. Note that the question that you are now referring to proves the reverse implication under additional conditions. – Martin R Oct 30 '20 at 21:38

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