Dirichlet's Test for Uniform Convergence

Theorem
Let $D$ be a set.

Let $\left({V, \left\vert{\,\cdot\,}\right\vert}\right)$ be a normed vector space.

Let $a_i, b_i$ be mappings from $D \to M$.

Let the following conditions be satisfied:


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


 * $(2): \quad \left\langle{b_n \left({x}\right)}\right\rangle$ be monotonic for each $x \in D$.


 * $(3): \quad b_n \left({x}\right) \to 0$ converge uniformly on $D$.

Then:


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

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

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

converges uniformly on $D$.

To do this we show that the Cauchy criterion holds.

Assign $\epsilon < 0$.

Then by definition of uniform convergence:
 * $\displaystyle \exists N \in \N: \forall x \in D: \forall n \ge N: \left\vert{b_n \left({x}\right)}\right\vert < \frac \epsilon 2$

Let $x \in D$ and $n > m \ge N$.

Then: