Does $f_n(x)=\frac{x}{n}\log(\frac{x}{n})$ converge uniformly in $(0,1)$? $f_n(x)=\frac{x}{n}\log\frac{x}{n}$
I tried to expand the bounds to $[0,1]$ and prove that the hypotheses of Dini's uniform convergence criterion are satisfied, but I'm not even sure I can expand the bounds since $f_n(0)$ isn't defined.

Hint:
$$
f_n(x)=\frac{x\log x}{n}-\frac{\log n}{n}\,x
$$
and $\lim_{x\to0^+}x\log x=0$.
