Let$\displaystyle (f_n)^\infty_{n=1}$ a sequence of function which defined recursively on interval $\displaystyle [a,b] $:

$\displaystyle f_n(x)=\sqrt{xf_{n-1}(x)}$, $\displaystyle f_0(x)\equiv1$

1. Prove that the sequence $\displaystyle (f_n)^\infty_{n=1}$ converges on interval $\displaystyle [a,b] $ to limit continuous function$\displaystyle f$.

2. Prove that the convergence is uniform.