Proof of alternating series test

**Stonehambey** · June 9th 2011, 12:39 PM

Hi,

I'm having trouble understanding the first line of Knapp's proof of the alternating series test. The Theorem states that

If for each $x$ in a nonempty set S, $\{ a_n(x)\}_{n \geq 1}$ is a monotone decreasing sequence of nonnegative real numbers such that $\lim_n a_n(x) = 0$ uniformly in $x$ , then $\sum^{\infty}_{n=1}(-1)^n a_n(x)$ converges uniformly.

His opening line in the proof is

The hypotheses are such that $\left| \sum^{n}_{k=m}(-1)^k a_k(x) \right| \leq \sup_x |a_m(x)|$ whenever $n \geq m$ ...

It's the above line I'm totally confused by. Can someone help explain to me how the hypothesis implies the above?

Thanks

**girdav** · June 9th 2011, 01:25 PM

I won't give the complete details but the idea is the following. We have
$0\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}\leq a_m(x)$ and $à\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}-a_{m+3}(x)\leq a_m(x)-a_{m+3}(x)\leq a_m(x).$

**Stonehambey** · June 9th 2011, 01:33 PM

Originally Posted by girdav

I won't give the complete details but the idea is the following. We have
$0\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}\leq a_m(x)$ and $a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}-a_{m+3}(x)\leq a_m(x)-a_{m+3}(x)\leq 0.$