Prove that this type of alternating series admits this supremum.

Question

Let $a_k\geq 0$ be a decreasing to zero numerical sequence. How how can we prove this inequality ? $$ \left|\sum_{k=n+1}^{\infty} (-1)^k a_k\right| \leq |a_n|$$

It may have something to do with Dirichlet's test or with Leibniz criterion, but I didn't find how to show it.

Answer 1

Let $b_i = a_i - a_{i+1}$ for all natural number $i$.

Then $b_i >0$, and $$a_n = \sum\limits_{i=n}^\infty b_i.$$

$$ \begin{split} |a_{n+1}-a_{n+2}+a_{n+3}-a_{n+4}+\ldots| & = |b_{n+1} + b_{n+3} + \ldots| \\ & = b_{n+1} + b_{n+3} + \ldots \\ & < b_n + b_{n+1} + b_{n+2}+ b_{n+3} + \ldots = a_n \end{split} $$