4 a. (10) Let $E \subset \mathbb{R}$ and suppose that $f_n \to f$ uniformly on $E$, as $n \to \infty$. If each $f_n$ is continuous, then $f$ is continuous.

b. (10) Let $f_n(x) = x^n$, $0 \leqslant x \leqslant 1$. Does $f_n$ converge uniformly?. What if $0 \leqslant x < 1$?