Proof of alternating series test

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June 9th 2011, 12:39 PM
Stonehambey

Proof of alternating series test

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 :)
June 9th 2011, 01:25 PM
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 $à\leq a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}-a_{m+3}(x)\leq a_m(x)-a_{m+3}(x)\leq a_m(x).$
June 9th 2011, 01:33 PM
Stonehambey

Quote:

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 $a_m(x)\underbrace{-a_{m+1}(x)+a_{m+2}(x)}_{\leq 0}-a_{m+3}(x)\leq a_m(x)-a_{m+3}(x)\leq 0.$

If $a_n(x)$ is nonnegative and monotone decreasing, how is $a_m(x)-a_{m+3}(x)\leq 0$ ?
June 9th 2011, 01:38 PM
girdav

Sorry, the second inequality is wrong, I meant $a_m(x)$ instead of 0. I will correct it.
June 9th 2011, 01:53 PM
Stonehambey

Ok, I think I see it now; I'll work on writing the argument out in full. Thanks