Let $X$ be a compact subset of $\mathbb{R}$. Let $f_n:X\to (0,\infty)$ be a sequence of function converging uniformly to a function $f:X\to(0,\infty)$. Show that the sequence of functions $\left(\frac{1}{n}\log(f_n)\right)$ converges uniformly to the identically zero function. Hint: Use the concavity of the logarithm function.
I wrote down the assumptions and conclusions using definition of uniform convergence, but I'm really stuck on them. Also I don't know where to use the concavity of the log.
The thing I know is that $\left(\frac{1}{n}\log(f_n)\right)$ converges pointwise to $0$, because $\frac{1}{n}\to0$ and $\log(f_n(x))\to\log(f(x))$ is bounded. Maybe I should use Dini's theorem, but I must show that each sequence $\left(\frac{1}{n}\log(f_n(x))\right)$ is monotone. Any hint will be appreciated.
Without any continuity assumptions this is false. Let $X=[0,1]$ and $f_n(x)=f(x)$ and $f(x)=\frac 1 x$ for $x>0, f(0)=1$. Then $\frac 1 n \log(f_n(x))$ does not tend to $0$ uniformly.