How to show that $p_n$ converges uniformly to $\sin x$ Let $p_n(x)$ be a sequence of polynomial with constant term $0$ and $p_n^{'}(x)$ converges to $\cos x$ uniformly on $(-\pi,\pi)$, we need to to show that $p_n(x)$ convrges uniformly to $\sin x$ on the same interval.
please first give hint, I want to try, if I can not then please give me detail solution. thank you.

$$\int_{0}^x p'_n(u)du=p_n(x)-p_n(0)=p_n(x)$$
Now try to use uniform convergence to show that:
$$\lim_{n\rightarrow\infty}\int_{0}^x p'_n(u)du=\int_{0}^x \lim_{n\rightarrow\infty}p'_n(u)du$$
