If $\sum a_{2k}$ exists then $\sum a_m$ exists?

Question

Let $\{a_0,a_1,a_2...\}$ be a sequence of real numbers let $s_n=\sum a_{2k}$. If $\lim_{n\rightarrow \infty} s_n $ exists then $\sum a_m$ exists. Is it true?

I don§t find a counter example

Answer 1

This is false. Let $$ a_n = \begin{cases}0 \text{, if } n \text{ is even} \\ 1 \text{, otherwise} \end{cases} $$

Then $\lim_{n \rightarrow \infty} s_n = 0$, but $\sum_{m=1}^\infty a_m = \infty$.

Answer 2

Let:

$$\begin{cases} a_m = 0 \text{ if m is even} \\ a_m = 1 \text{ if m is odd }\end{cases}$$ Then $\sum a_{2k}=0$ and $\sum a_m =\infty$.