Proving $\sum_{k=1}^{\infty}(-1) \ ^{ k+1} ak <\infty$ for $a_n\downarrow 0$. Let $(a_n)$ be a sequence of positive real numbers such that for each $n\in\Bbb N$, $a_{n+1}\le a_n$, and $\lim\limits_{n\to\infty}a_n=0$. Prove that the following series is convergent: $$\sum_{k=1}^{\infty} (-1) \ ^{ k+1} a_k $$ I started by thinking that there is $n_0$ s.t for every $n>n_0$, $a_n <1$ and then I thought , using compassing to geometric progression with $q = a_{n_0}$. Am I in the right direction ?

**Hint**

Denote $(S_n)$ the given sequence (partial sum). Prove that the two sequences $(S_{2n})$ and $(S_{2n+1})$ are adjacent. Conclude.