$I_n = \int_{0}^{\frac{\pi}{2}}(\cos t)^n \ dt$ converges to 0? How one can prove that the sequence $\left ( I_n \right )$ defined as $$ I_n = \int_{0}^{\frac{\pi}{2}}(\cos t)^n  \ dt, $$ $n \in \{ 0,1,2,...\}$ converges to $0$?
Is easy to show, by the way, that the sequence is decreasing because, for $t  \in  (0, \pi/2)$, $$(\cos t)^{n+1}<(\cos t)^n \Rightarrow I_{n+1} < I_{n}, \ \forall n $$

HINT: $\forall\varepsilon>0 \exists\delta>0: \cos x<1-\varepsilon$ on $(\delta,\pi/2]$ and $\delta$ decreases to $0$ when $\varepsilon$ tends to $0$.
