Let $f_n:X_n\rightarrow W$ be continuous, $g_n:X_n\rightarrow W$ be a continuous map such that $W$ is complete in a complete metric space $Y$ and $X_n$ is compact. Assume the following is given: For every $\epsilon>0$ there exists a natural number $N$ such that if $n\geq N$ and $x_n\in X_n$ then $d(f_n(x_n),g_n(x_n))<\epsilon$ and $\xi:W\rightarrow Y$ is a continuous map.
I would like to know if the following holds:
For every $\epsilon>0$ there exists a natural number $N'$ such that for $n\geq N'$ if $x_n\in X_n$ then $d(\xi(f_n(x_n)),\xi(g_n(x_n))<\epsilon$
Take $X_n = [0,1]$ and $W=Y = \mathbb R$. Assume you can choose $f_n,g_n,x_n$ such that $$ f_n(x_n) = n+ \frac 1 n,\quad g_n(x_n) = n. $$ Then put $\xi(x) = x^2$ to get a counter example: $$ d(\xi(f_n(x_n)) , \xi(g_n(x_n))) = \left( n+ \frac 1 n \right)^2 - n^2 = 2 + \frac{1}{n^2} \geq 2. $$ Now you are left to build such $f_n,g_n,x_n$. You can take $f_n$ constant to $n+ \frac 1 n$, $g_n$ constant to $n$ and $x_n$ any point of $[0,1]$.