>>11306566So if f is point wise convergent, then for every epsilon we get N(x) s.t. at x, that f(x)-f_{N(x)}(x) < epsilon where epsilon is given. Choose the largest such N(x) and let that be N, and since g(x) = f(x)-f_{N}(x) is continuous, we have that there is delta s.t. for d(x,y)<delta that g(x)-g(y) < 2epsilon. Since this holds for all x, and such a construction can be made for all epsilon, it holds that f_{n}(x) is uniformly convergent.
