Let $\displaystyle (a_n)$ be a sequence.

(i) Prove that if $\displaystyle \sum_{n=1}^{\infty}{a_n}$ converges, then $\displaystyle \sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})}$ also converges.

(ii) Prove that if $\displaystyle \sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})}$ converges and $\displaystyle a_n \rightarrow {0}$, then $\displaystyle \sum_{n=1}^{\infty}{a_n}$ converges.

For (i), can i use the concept of subsequences to tackle this question.

For (ii), how do I start this question?

Thanks in advance!