Abel's Test for Uniform Convergence

Theorem
Let $\sequence {\map {a_n} z}$ and $\sequence {\map {b_n} z}$ be sequences of complex functions on a compact set $K$.

Let $\sequence {\map {a_n} z}$ be such that:
 * $\sequence {\map {a_n} z}$ is bounded in $K$
 * $\displaystyle \sum \size {\map {a_n} z - \map {a_{n + 1} } z}$ is convergent with a sum which is bounded in $K$
 * $\displaystyle \sum \map {b_n} z$ is uniformly convergent in $K$.

Then $\displaystyle \sum \map {a_n} z \map {b_n} z$ is uniformly convergent on $K$.