Dirichlet's Test for Uniform Convergence

Theorem
Suppose:


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


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


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

Then:


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

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

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

To do this we show that the Cauchy Criterion holds.

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

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