Dirichlet's Test for Uniform Convergence

Theorem
Suppose: 1. The partial sums of $$\sum_{n=1}^{\infty}a_n(x)$$ are uniformly bounded on $$D$$. 2. $${b_n(x)}$$ is monotonic for each $$x\in D$$. 3. $$b_n(x)\rightarrow 0$$ uniformly on $$D$$. Then: $$\sum_{n=1}^{\infty}a_n(x)b_n(x)$$ 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 $$\sum_{n=1}^{\infty}|b_n(x)-b_{n+1}(x)|$$ converges uniformly on $$D$$. To do this we show that the Cauchy Criterion holds.

Assign $$\epsilon<0$$, then there exists a natural number $$N$$ such that $$\forall x\in D$$ and $$\forall n\geq N, |b_n(x)|<\frac{\epsilon}{2}$$.

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

Q.E.D.