I came up with a solution to a simple problem that I was working on but it seemed too simplistic so I doubt that my solution is correct which is why I want some help. $f_n(x)$ is a sequence of complex valued functions on set $X$ and $f(x) $ is another complex function on the same set.
Show that $f_n(x)$ converges to $f(x)$ uniformly for $n \rightarrow \infty$ if: $$ \sup_{x\in X}|f_n(x)-f(x)| \rightarrow 0. $$ My attempt: $$ \inf_{x \in X}|f_n(x)-f(x)|≤ |f_n(x)-f(x)| ≤ \sup_{x \in X} |f_n(x)-f(x)|$$ $$ \lim_{n \rightarrow \infty} \inf_{x \in X}|f_n(x)-f(x)|≤\lim_{n \rightarrow \infty} |f_n(x)-f(x)|≤ \lim_{n \rightarrow \infty} \sup_{x \in X}|f_n(x)-f(x)| $$ $$ \lim_{n \rightarrow \infty} |f_n(x)-f(x)| ≤\lim_{n \rightarrow \infty} \sup_{x \in X}|f_n(x)-f(x)|=0 ≤ \epsilon.$$ Initially I wanted to make an argument using the squeeze theorem for limits but I didn't know how to proceed with the limit of infimum so this was what I had. I don't think its entirely wrong but I'm sure I'm missing something here.