 Solution

The closed bounded set $A\subset {\mathbb{R}}^m$ is compact by the Heine-Borel theorem. Choose a convergent subsequence $x_{n_j}\to x_0$. For fixed $N$ and sufficiently large $j$, monotonicity givesContinuity and passage to the limit give $g_N(x_0)\ge \delta$ for every $N$, hence ${\lim }_N{g}_N(x_0)\ge \delta$.

For the deduction, set $g_n=f_n-f$. These functions are continuous, decrease pointwise, and tend to zero. If convergence were not uniform, some $\delta >0$ would admit $x_n$ with $g_n(x_n)\ge \delta$. The result just proved would produce $x_0$ with $\lim g_n(x_0)\ge \delta$, contradicting pointwise convergence. Therefore $f_n\to f$ uniformly. This is Dini's theorem.