Dirichlet's Test for Uniform Convergence

Theorem
Let $D$ be a set.

Let $\struct {V, \norm {\,\cdot\,} }$ 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 $\displaystyle \sum_{n \mathop = 1}^\infty \map {a_n} x$ be bounded on $D$


 * $(2): \quad \sequence {\map {b_n} x}$ be monotonic for each $x \in D$


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

Then:


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

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

All we need to show is that:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \size {\map {b_n} x - \map {b_{n + 1} } x}$

converges uniformly on $D$.

To do this we show that the Cauchy criterion holds.

Assign $\epsilon < 0$.

Then by definition of uniform convergence:
 * $\displaystyle \exists N \in \N: \forall x \in D: \forall n \ge N: \size {\map {b_n} x} < \frac \epsilon 2$

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

Then: