Let $f_{n},g_{n}:[0,1]\rightarrow\mathbb R$ be given by $g_{n}(x)=\frac{2nx}{1+n^3x^2}$ and $f_{n}(x)=\frac{\log(1+n^3x^2)}{n^2}$
Show that $g_{n}(x)$ converges uniformly to $0$. (Completed)
Now use this this to show that $f_{n}(x)$ also converges uniformly to $0$
Obviously I need to make use of $f_{n}(x)=\frac{\log(2nx)-\log(g_{n}(x))}{n^2}$, just not sure where to go from here.
Regarding using $f_n$, here's a hint. Note that $g_n=f_n'$. We have $$ |f_n(x)| = \left|f(0) + \int_0^x g_n(y)dy\right|\le |f(0)| + \int _0^1 |g_n(y)| dy $$
Though the other answer was pretty simple