Suppose there exists a sequence of functions {gn}n=1∞ uniformly converging to 0 on A. Now suppose we have a sequence of functions {fn}n=1∞ and a function f on A such that
∣fn(x)−f(x)∣≤gn(x)
for all x∈A. Show that {fn}n=1∞ converges uniformly to f on A.