i was wondering if i might be able to get help with the following question:
Let $(X,d)$ and $(Y,\rho)$ be metric spaces with $X$ compact. Let $f_n$ and $f$ be continuous functions from $X$ to $Y$ and suppose that $f_n\to f$ pointwise as $n\to\infty$. Show that if there is a constant $N$ such that $$ \rho(f(x),f_n(x)) \le N \rho(f(x),f_m(x)) \quad\mbox {for all $m,n$ natural numbers with $n\ge m$,} $$ then $f_n\to f$ uniformly.