Let f,g: [0,1] ￫ ℝ be two continuos functions. Prove that
lim ∫{x=0,x=1} f(x^n)g(x)dx = f(0) ∫{x=0,x=1} g(x)dx.
n￫∞
The things inside the brackets are the limits of the integrals
Just prove that
$\displaystyle \left|\int_0^1 f(x^n)\cdot g(x)\, dx-f(0)\cdot\int_0^1 g(x)\, dx\right|\rightarrow 0, \quad \text{ as } n\rightarrow \infty$
or, maybe better still, that
$\displaystyle \left|\int_0^1 \Big(f(x^n)-f(0)\Big)\cdot g(x)\, dx\right|\rightarrow 0, \quad \text{ as } n\rightarrow \infty$
Because, consider:
(1) f being continuous implies that for any $\displaystyle \varepsilon>0$ there exists a $\displaystyle \delta >0$ such that $\displaystyle |f(x)-f(0)|<\varepsilon$ if $\displaystyle 0\leq x< \delta$.
(2) the larger n the longer does $\displaystyle x^n$ stay close to 0 (for example, smaller than $\displaystyle \delta$) as you move x from 0 to 1.
(3) f and g being continuous in [0;1] also implies that $\displaystyle |f(x^n)\cdot g(x)|\leq M$ for some constant M and all $\displaystyle x\in [0;1]$.