Let $f_n,f,g_{m,n}$ and $g_m$ be real valued functions where $m,n\in\mathbb{N}$.
Suppose that as $n\to\infty$:
$f_n$ converges to $f$ uniformly.
$g_{m,n}$ converges to $g_m$ uniformly.
Also suppose that as $m\to\infty$:
- $g_{m,n}$ converges to $f_n$ uniformly.
My question: Does $g_{m}$ converge to $f$ uniformly as $m\to\infty$?
Edit: Using @TonyS.F.'s suggestion. From the triangle inequality $$ \|f - g_m\|_\infty \leq \|f - f_n\|_\infty + \|f_n - g_{m.n}\|_\infty + \|g_{m.n} - g_m\|_\infty $$ for all $m,n\in\mathbb{N}$. Therefore, $$ \lim_{m\to\infty} \|f - g_m\|_\infty \leq \lim_{m\to\infty} \lim_{n\to\infty} \|f - f_n\|_\infty + \|f_n - g_{m.n}\|_\infty + \|g_{m.n} - g_m\|_\infty $$ and so $g_m$ converges to $f$ uniformly. QED.