Fact:"If limit of a real sequence $\{a_n\}_{n \in \mathbb{N}}$ exists then the limit of all subsequences of $\{a_n\}_{n \in \mathbb{N}}$ exist and is equal to the limit of the sequence $\{a_n\}_{n \in \mathbb{N}}$."
Question:Does the same hold in the case of a series,in the sense that "If $\sum\limits_{n=0}^{\infty}a_n$ is a series of real numbers whose sum exists finitely or infinitely,then can we say that the sum $\sum\limits_{n=0}^{\infty}a_{2n}$ exists finitely or infinitely?
I tried it by considering the sequence of partial sums of the series $\sum\limits_{n=0}^{\infty}a_n$ and $\sum\limits_{n=0}^{\infty}a_{2n}$,in a hope that I can apply the above fact,but as the sequence of partial sums of the series $\sum\limits_{n=0}^{\infty}a_{2n}$ is not subsequence of the sequence of the partial sums of the series $\sum\limits_{n=0}^{\infty}a_n$ hence the above fact can not be applied.
The above statement may or my not be true.Kindly someone help me.