let (gn) be a sequence of functions on D such that gn+1(x)< gn(x) for all x in D. if (gn)-> 0 uniformly on D, show that Σ (-1)^n(gn) converges uniformly on D
The series is pointwise convergence by the alternating series test (since the series is decreasing all the terms are non-negative on $\displaystyle D$ etc).
The conditions imply that there exists a decreasing positive sequence $\displaystyle a_n \to 0$ as $\displaystyle n\to \infty$ such that $\displaystyle g_n(x)\le a_n$ for all $\displaystyle x \in D$.
Then (since the remainder for a truncated alternating series is bounded by the absolute value of the first neglected term):
$\displaystyle \left| \left(\sum_0^{\infty} (-1)^n g_n(x)\right) - \left(\sum_0^n (-1)^n g_n(x)\right) \right|\le g_{n+1}(x) \le a_{n+1}$
etc.
CB