The problem
Assume $f\in\mathscr{R}(\alpha)$ on $[a,b]$, and prove that there are polynomials $P_n$ such that $$ \lim\limits_{n\rightarrow\infty}\int^b_a\left|f-P_n\right|^2\,d\alpha=0\mbox{.} $$
Some results from the book I shall use.
Ex 6.12 Suppose $f\in\mathscr{R}(\alpha)$ and $\epsilon>0$. Prove that there exists a continuous function $g$ on $[a,b]$ such that $\left\{\int^b_a\left|f-g\right|^2\,d\alpha\right\}^{1/2}<\epsilon$.
Ex 7.2 If $\{f_n\}$ and $\{g_n\}$ converge uniformly on a set $E$, prove that $\{f_n+g_n\}$ converges uniformly on $E$. If, in addition, $\{f_n\},\{g_n\}$ are sequences of bounded functions, prove that $\{f_ng_n\}$ converges uniformly on $E$.
Thm 7.26 If $f$ is a continuous real function on $[a,b]$, there exists a sequence of polynomials $P_n$ such that $$ \lim\limits_{n\rightarrow\infty}P_n(x)=f(x) $$ uniformly on $[a,b]$.
Proof of the problem. By Ex 6.12, for each positive integer $n$, we can find a continuous function $g_n$ on $[a,b]$ such that $$ \int^b_a\left|f-g_n\right|^2\,d\alpha<\frac{1}{n}\mbox{.} $$ By Thm 7.26, there is a sequence $\{P^{(n)}_k\}$ of polynomials converging uniformly to $g_n$. By Ex 7.2, $$ \lvert f-P^{(n)}_k\rvert^2\rightarrow\lvert f-g_n\rvert^2\quad\mbox{uniformly}$$ as $k\rightarrow\infty$. Therefore, there exists an integer $K(n)$ such that $$ \left|\int^b_a\lvert f-P^{(n)}_k\rvert^2\,d\alpha-\int^b_a\lvert f-g_n\rvert^2\,d\alpha\right|<\frac{1}{n} $$ provided only that $k\geq K(n)$; hence $$ \int^b_a\lvert f-P^{(n)}_{K(n)}\rvert^2\,d\alpha<\frac{2}{n} \mbox{.} $$ Define a sequence $\{P_n\}$ of polynomials by $P_n=P^{(n)}_{K(n)}$. It is clear from the last inequality that $$ \int^b_a\left|f-P_n\right|^2\,d\alpha\rightarrow0 \mbox{.} $$ $\tag*{$\blacksquare$}$
Does this look correct?