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 etc).
The conditions imply that there exists a decreasing positive sequence as such that for all .
Then (since the remainder for a truncated alternating series is bounded by the absolute value of the first neglected term):
etc.
CB