Suppose that $g_n \geqslant 0$ is a sequence of integrable functions which satisfies $\lim\limits_{n \to \infty} \int_a^b g_n(x) \mathrm{d} x = 0$.
a) Show that if $f$ is an integrable function on $[a,b]$, then $\lim_{n\to \infty} \int_a^b f(x) g_n(x) \mathrm{d}x = 0$.
b) Prove that if $f$ is integrable on $[0,1]$, then $\lim_{n\to \infty} \int_0^1 x^n f(x) \mathrm{d}x = 0$.
Here is what I have so far; We are given that $g_n \geqslant 0$, with $\lim_{n\to\infty}\int^b_ag_n(x)=0$. We also have that $f$ is integrable on $[a,b]$. Then, by general form of Mean Value Theorem for Integrals (which I already proved in another problem), we know $\exists c\in[a,b]$ such that $\int^b_af(x)g_n(x)dx=f(c)\int^b_ag_n(x)dx$. Thus we have: $\lim_{n\to\infty}\int^b_af(x)g_n(x)dx\Rightarrow \lim_{n\to\infty}f(c)\int^b_ag_n(x)dx\Rightarrow f(c)\lim_{n\to\infty}\int^b_ag_n(x)\Rightarrow f(c)⋅0=0. $ (b) We are given that $f$ is integrable on $[0,1]$. Let $g_n(x)=x^n$. Our claim, is that $g_n(x)\to g(x)=\left\{\begin{array}{ccc}0&,&x\ne1\\1&,&x=1\end{array}\right.$ on $[0,1]$. Then let $x_0\in [0,1)$. Then $\lim_{n\to\infty}x^n_0=0$. Thus $g_n(x)\to g(x)$ pointwise on $[0,1]$. Further $\lim_{n\to\infty}\int^b_ag_n(x)=\int^b_ag(x)=0$. Thus, by (a), we have that $\lim_{n\to\infty}\int^b_ax^nf(x)dx=0$.
Is this correct?