Originally Posted by

**paupsers** The problem states:

Prove that the following limit exists and find the limit.

$\displaystyle lim_{n\rightarrow\infty}\int_0^{\pi/2}1-\frac{\sqrt{x\sin {nx}}}{n}dx$

I've already shown that the pointwise limit is 1, but when I try to show that it converges uniformly using the definition that

If $\displaystyle |f_n(x)-f(x)|<\epsilon$

then it's uniformly convergent, I run into a slight problem... I get to a point where I have

$\displaystyle |\frac{\sqrt x\sqrt{\sin nx}}{n}|$ and I want to show (after a few more inequalities) that this is less than epsilon.

However, how can I account for the fact that sometimes the $\displaystyle \sqrt{\sin nx}$ is negative, thus giving me a nonreal value?

What this all boils down to, is can I say that

$\displaystyle |\frac{\sqrt x\sqrt{\sin nx}}{n}|\leq|\frac{\sqrt x}{n}|$