Dini's theorem:
if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform.
What's the meaning of monotone sequence of functions?
How can a sequence of functions be monotone?
If I have a sequence $f_n(x)$, this means that $f_1(x) \le f_2(x) \le f_3(x) \le ... \le f_n(x)$ (Assume increasing) for every $x$?
Edit: Does that mean that for all $x$, $f_1(x) \le f_2(x_1)$ for any $x_1$ or that for all $x$, we fix $x$ and then $f_1(x) \le f_2(x)$?
There is probably a definiton out there, I just couldn't find. Please clear this up for me.
You have it right: the sequence $\{f_n\}$ is increasing if $f_n(x)\leq f_{n+1}(x)$ for all $x$ and all $n$. It is decreasing if $f_n(x)\geq f_{n+1}(x)$ for all $x$ and all $n$, and monotone if it is increasing or decreasing.