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
or, maybe better still, that
Because, consider:
(1) f being continuous implies that for any there exists a such that if .
(2) the larger n the longer does stay close to 0 (for example, smaller than ) as you move x from 0 to 1.
(3) f and g being continuous in [0;1] also implies that for some constant M and all .