1. ## need help

I need help in the following questions :

Qn1 : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is a ∈ [0, 1] such that f(g(a)) = g(f (a)).

Qn2 : Compute the limit lim(n→∞) integrate (0,pi,cos(x^n))

First observe that $g\circ f$ and $f\circ g$ are onto since $f$ and $g$ are.
Suppose that $g(f(x))>f(g(x))$ for all $x\in[0,1]$. This means that $g(f(x_0))> 1$ for some $x_0\in[0,1]$. And $g(f(x))< f(g(x))$ for all $x$ implies there exists a $x_0\in [0,1]$ such that $g(f(x_0)) < 0$.
(You can use intermediate value theorem to show: $f(g(x))-g(f(x)) < 0$ and $f(g(x))-g(f(x))> 0$ for some $x$ implies there exists a $a\in [0,1]$ such that $f(g(a))-g(f(a)) = 0$, by continuity of $f(g(x))-g(f(x))$).