Dirichlet's Test for Uniform Convergence

Theorem
$\newcommand{\abs} [1] {\left| {#1} \right|}$ Suppose:


 * The partial sums of $\displaystyle \sum_{n=1}^\infty a_n (x)$ are bounded on $D$.


 * ${b_n(x)}$ is monotonic for each $x\in D$.


 * $b_n(x)\to 0$ converges uniformly on $D$.

Then:


 * $\displaystyle \sum_{n=1}^{\infty}a_n(x)b_n(x)$ converges uniformly on $D$.

Proof
Suppose $b_n(x)\geq b_{n+1}(x)$ for each $x \in D$.

All we need to show is that $\displaystyle \sum_{n=1}^{\infty}|b_n(x)-b_{n+1}(x)|$ converges uniformly on $D$.

To do this we show that the Cauchy Criterion holds.

Assign $\epsilon<0$, then $\exists N \in \N$ such that $\displaystyle \forall x \in D, \forall n \ge N: \abs{b_n(x)}< \frac \epsilon 2$.

If $x\in D$ and $n > m \ge N$ then,