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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.

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    Looks like a good definition e.g $f_n(x) = \frac{x}{n}$ defines a monotone sequence of functions on [0,1]. – 2017-01-04
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    @Paul So in my edit, the second definition (we fix $x$) is the right one, right? – 2017-01-04
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    Yes, it is a pointwise definition and Dini's theorem gives a strong result of uniform convergence. – 2017-01-04

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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.

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    For all $n$ or for every $n > N$? Thanks for the answer. – 2017-01-04
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    Usually for all $n$, but it could be adjusted to all $n>N$ for some fixed $N$ without changing much. – 2017-01-04
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    So I can use Dini's throrem with some fixed $N$? Thing is in Wikipedia they say it needs to be for all $n$. – 2017-01-04
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    I edited my question. I would really appreciate if you take a quick look :) – 2017-01-04
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    The requirement is that for each $x$, $f_n(x)\leq f_{n+1}(x)$ for all $n$ (if the sequence is increasing). It's not too hard to adjust the usual Dini's theorem to the case where the sequence is only increasing for $n>N$; just define a new sequence by $g_n=f_n$ if $n>N$, and $g_n=f_N$ if $n\leq N$. – 2017-01-04
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    This new sequence is increasing, and has the same limit as the original one. – 2017-01-04