Calculate $\lim\limits_{n \to \infty}\int\limits_{0}^{1} \frac{f_n(x)}{\sqrt{|x-\frac{1}{n}|}}$ given that $|f_n(x)|\leq 1$ and $f_n \to f$ pointwise.

Question

Calculate $\lim\limits_{n \to \infty}\int\limits_0^1 \frac{f_n(x)}{\sqrt{|x-\frac{1}{n}|}}$ given that $|f_n(x)|\leq 1$ and $f_n \to f$ pointwise.

The way I did this problem is very messy so I was hoping someone could point me to a simpler solution.

How I did it is I split up th integral into two parts $[0,1/n],[1/n,1]$ then on each individually I showed that $\frac{1}{\sqrt{|x-1/n|}}$ converges as desiered. Then used general DCT on both parts to show convergence to $\int_{0}^{1}\frac{f(x)}{\sqrt{x}}$. I am pretty sure there is a nicer solution and i am just over complicating things

Answer 1

Let $g_n(x) = \frac{f_n(x)}{\sqrt{|x-1/n|}}$ and $g(x) = \frac{f(x)}{\sqrt{|x|}}$. If $f_n \to f$ point wise, then $g_n \to g$ pointwise on $(0,1)$. Since $|f_n| \leq 1$, we must also have $|g_n(x)| \leq \frac{1}{|\sqrt{x}|}$. Since $\int_0^1 \frac{1}{\sqrt{x}} dx = 2$, you can apply Lebesgues Dominated Convergence theorem.