Suppose that $(f_n(x,y))_{n\in\mathbb{N}}$ is a sequence of real valued functions and that they converge to $f(x,y)$ uniformly on compacts sets. Suppose $x\in\mathbf{X}\subset\mathbb{R}$ and $\mathbf{X}$ is a finite set i.e. has a finite amount of members. Suppose $y\in\mathbf{Y}\subseteq\mathbb{R}$.
Does $g_n(y) := \max_{x\in\mathbf{X}} f_n(x,y)$ converge uniformly on compact sets to $g(y) :=\max_{x\in\mathbf{X}} f(x,y)$ as $n\to\infty$?
My attempt: Let $\mathbf{K}_x \subseteq \mathbf{X}$ and $\mathbf{K}_y \subseteq \mathbf{Y}$ be compact sets. We know that on $\mathbf{K}_x \times \mathbf{K}_y$ that $f_N$ converges to $f$ uniformly. Then is this enough to conclude that $g_n$ converges to $g$ on $\mathbf K_x \times \mathbf{K}_y$? Did I make a jump in logic?
We can essentially do this by induction on $|X| = n$ starting from the case $n=2$ and using the identity $$ \max(f,g) = \frac{f+g}{2} + \frac{|f-g|}{2}, $$ so we see that taking the max of two functions is a continuous operation in the sense that if $f_n\to f$ and $g_n\to g$, then $\max(f_n,g_n)\to\max(f,g)$. Taking the max of $n\ge 2$ functions is the same as taking $\max(\max(f_1,\dots,f_{n-1}),f_n)$.