Problem

請寫出計算及證明過程，勿只寫答案。

3. Let $f_0(x)$ be a continuous function on $[0,2\pi]$. Define a sequence of functions on $[0,2\pi]$ by

$$f_n(x)=\int_0^x f_{n-1}(s)\sin(s)\,ds.$$

(a) For every $x\in[0,2\pi]$, determine $\lim_{n\to\infty}f_n(x)$.

(b) Is the convergence in part (a) uniform on $[0,2\pi]$? Justify your answer.