How can we fill in some missing details in this proof?

Question

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\to\infty$. Show that $f_n$ converges uniformly to $f$.

My working:

We can apply the Dini's Theorem as follows:

WLOG let $(f_n)$ be monotonically increasing sequence of continuous functions. Since for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\to\infty$, then $f_n(x)\to f(x)$ for every $x\in X$ and hence $f_n\to f$ pointwise. Since $f$ is continuous. Then by Dini's Theorem $f_n\to f$ uniformly.

But I have some doubts in my proof:

1. What if $(f_n)$ is neither monotonically increasing nor decreasing, is it possible if $f_n$ is alternating but $\rho(f_n(x),f(x))$ still decreases to $0$?
2. I am not sure how can $\rho(f_n(x),f(x))$ decreases to $0$ implies $f_n\to f$ pointwise. I did not write the details in my proof because I am not sure whether or not $\rho(f_n(x),f(x))$ decreases to $0$ really implies $f_n\to f$ pointwise. If yes, how can we fill the details?
3. Is my overall proof correct? If not how can we fix it (by still using Dini's Theorem)?

Thanks for the help!

Answer 1

For all $x\in X$, we have $\rho(f_n(x),f(x))\geqslant \rho(f_{n+1}(x),f(x))$. It follows that $$\sup_{x\in X}\rho(f_n(x),f(x))\geqslant\sup_{x\in X} \rho(f_{n+1}(x),f(x)).$$ Since $\lim_{n\to\infty} \rho(f_n(x),f(x))=0$ for all $x\in X$ we conclude that $$\lim_{n\to\infty}\sup_{x\in X} \rho(f_n(x),f(x))=0,$$ and hence $f_n$ converges uniformly.