Let $(X,d)$ and $(Y,\rho)$ be metric spaces, $X$ compact, and suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that $\rho(f_n(x),f(x))$ decreases to 0 as $n\to\infty$ for each $x\in X$. Prove that $f_n$ converges uniformly to $f$.
I could start doing this problem by writing some definitions:
Since $X$ is compact, then every sequence from $X$ has a subsequence which converges to an element of $X$. Since $f$ and $f_n$ are continuous, then $Y$ is also compact. We need to show that for every $\epsilon>0$, there exists $N$ such that $n\geq N\implies\rho(f_n(x),f(x))<\epsilon,\forall x\in X$.
Some of my doubts are:
Does the meaning of the sentence $\rho(f_n(x),f(x))$ decreases to $0$ as $n\to\infty$ for each $x\in X$ means $\rho(f_n(x),f(x))\to0$ as $n\to\infty$ for each $x\in X$?
If yes, then from definition of limit we have for every $\epsilon>0$, there exists $N$ such that $n\geq N\implies|\rho(f_n(x),f(x))-0|=|\rho(f_n(x),f(x))|=\rho(f_n(x),f(x))<\epsilon$. Then we are done.
Am I correct? I am not sure though.
Also I am not sure if we can use Dini's Theorem, if we can, how can we use it?
Could anybody please give some help? Thanks so much.