Originally Posted by

**HenryB** The question goes on to say:

"For each $\displaystyle \epsilon > 0$ find a subset $\displaystyle E_\epsilon$ of [0,1] such that $\displaystyle f_n(x) \rightarrow 0$ uniformly on $\displaystyle [0,1]$ \ $\displaystyle E_\epsilon$.

Now it's clearly not what the question is after (I don't actually know what the question is trying to say) but can't we just take $\displaystyle E_\epsilon = (0,1]$ for all $\displaystyle \epsilon > 0$?

Secondly, can we find a subsequence $\displaystyle ({g_n}_r)$ such that $\displaystyle lim_{r \rightarrow \infty} {g_n}_r(x) = 0$ almost everywhere?