Assume to the contrary. That is for some e>0 there is no N s.t for all x, n>=N |fn(x)-f(x)|<e . Thus we have |f_n(x)-f(x)|>= e for infinitely many n and some x. Consider the subsequence consisting of all such n (we will rename them m). We wish to show S|f_m(x)-f(x)| does not go to 0 as m goes to infinity. At some point x, |f_m(x)-f(x)|>=e for all m . Thus f_m(x) does not go to f(x).

Apr 30th, 2015

The above is not a proof. Just checking, do you really mean limit point or pointwise limit? I can definitely prove this with respect to the pointwise limit