# Show that $\lim_{n\rightarrow \infty}\int_{a}^{\pi/2}\sqrt{n} \cos^n(x)dx=0$ for any $a\in \left(0,\frac{\pi}{2}\right)$

I am trying to prove that $$\lim_{n\rightarrow \infty}\int_{a}^{\pi/2}\sqrt{n} \cos^n(x) dx=0\hspace{10mm}\forall a\in \left(0,\frac{\pi}{2}\right)$$.

My trial

I am thinking of using Dominated or monotone convergence theorem. Since the limit goes to $0$ from non zero value, $\{\sqrt{n}\cos^n\}$ should be somehow decreasing. Thus, the way seems to lean toward dominated convergence theorem.

But even if I can somehow put the limit inside of integral, I am not sure if $$\lim_{n\rightarrow \infty} \sqrt{n}\cos^n(x)=0 \hspace{4mm} a.e.$$ is true.

I hope I can get any hint from this site :) Thank you in advance!

You can just see that $\cos$ is decreasing from $a$ to $\frac{\pi}{2}$. So you have :
$$\int_a^{\pi/2} \sqrt{n}\cos(x)^n dx \leqslant \sqrt{n}\cos(a)^n (\pi/2 - a) \underset{n\rightarrow\infty}{\longrightarrow 0}$$
since $\cos(a) \in [0,1)$ and by compared growths.