I dont like your proof of (a). I would prefer
f_n(x) = (1/n) sin(nx),
-1/n <= f_n(x) <= 1/n
and as lim(n->infty) -1/n = lim(n->infty) 1/n =0, we conclude:
lim(n->infty) f_n(x) =0.
Captain, I would have done things your way under normal circumstances, but i was really tired when i was doing this stuff. the way i messed up the derivative attests to that.
Thanks for bringing me back to my senses. I'm going home to get some sleep now