= 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 gives
$$
g_N(x_{n_j})\geq g_{n_j}(x_{n_j})\geq\delta.
$$
Continuity and passage to the limit give $g_N(x_0)\geq\delta$ for every $N$, hence $\lim_Ng_N(x_0)\geq\delta$.
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 $g_n(x_n)\geq\delta$. The result just proved would produce $x_0$ with $\lim g_n(x_0)\geq\delta$, contradicting pointwise convergence. Therefore \b[$f_n\to f$ uniformly]. This is <Dini's theorem>.
Back to article page
