Sequences of functions Uniform Convergence Question

Question

Let g and h be positive and continuous on $[a,b]$ and define $f_n(x):=\frac{ng(x)}{1+n^2h(x)}$. Prove $n\sin{(f_{n})}\rightarrow\frac{g}{h}$ uniformly on $[a,b]$

Answer 1

We will use the $sin$ series. So we have $nsin(f_n(x))=n\cdot \sum_{k=0}^{\infty} \frac {(-1)^k}{(2k+1)!}\cdot (\frac {ng(x)}{1+n^2h(x)})^{2k+1}$ For $k=0$ we have the term $\frac {n^2g(x)}{1+n^2h(x)}\to \frac {g(x)}{h(x)}$ and all the other terms go to $0$.

Now you have to show that $\| nsin(f_n(x))-\frac {g(x)}{h(x)}\|_{\infty}\to 0$. You can try from here.