Exercise 7.4.9

Let $g_n$ and $g$ be uniformly bounded on $[0,1]$, meaning that there exists a single $M>0$ satisfying $|g(x)|\le M$ and $\left|g_n(x)\right|\le M$ for all $n\in N$ and $x\in [0,1]$. Assume $g_n\to g$ pointwise on $[0,1]$ and uniformly on any set of the form $[0,\alpha ]$, where $0<\alpha <1$. If all the functions are integrable, show that $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_0^1{g}_n={\mathrm{\int \; <!--\; nolimits\; -->}<!--\; FUNCTION\; APPLICATION\; -->}_0^{1}g$.