Uniform Convergence of a sequence

Question

I need to prove that the sequence $f_n(x) = (x/n)\log(x/n)$ converges uniformly to its limit function on the interval $(0,1)$ (log is the natural logarithm) . I tried to make an estimate but it didn't work.

Answer 1

Note that $\lim_{t \downarrow 0} t \log t = 0$.

Let $\epsilon>0$ and choose $\delta>0$ such that $|t \log t| < \epsilon$ for all $t \in (0,\delta)$. Now choose $N$ such that $\frac{1}{N} < \delta$. Then if $n \ge N$, we have $\frac{x}{n} \le \frac{1}{N} < \delta$, and so $|f_n(x)| < \epsilon$.

Answer 2

Hint: $\sqrt{t} \ln t \to 0$, $t \to 0$, hence for some $\epsilon > 0$ we have $\sqrt{t} \ln t < 1$ for $t < \epsilon$. Therefore $$\frac{x}{n} \ln \frac{x}{n} < \sqrt{\frac{x}{n}}<\sqrt{\frac{1}{n}}$$ for sufficiently large $n$.