If $f_n\colon [0, 1] \to [0, 1]$ are nondecreasing and $\{f_n\}$ converges pointwise to a continuous $f$, then the convergence is uniform Suppose that $\{f_n\}$ is a sequence of nondecreasing functions which map the unit interval into itself. Suppose that $$\lim_{n\rightarrow \infty} f_n(x)=f(x)$$ pointwise and that $f$ is a continuous function. Prove that $f_n(x) \rightarrow f(x)$ uniformly as $n \rightarrow \infty$, $0\leq x\leq1$. Note that the functions $f_n$ are not necessarily continuous.
This is one of the preliminary exam from UC Berkeley, the solution goes like this:
Because $f$ is continuous on $[0,1]$, which is compact, it is then uniformly continuous. Hence there exists $\delta >0$ such that if $|x-y|<\delta$ then $|f(x)-f(y)|<\epsilon$.
We then partition the interval with $x_0=0, \cdots ,x_m=1$ such that the distance $x_{i}-x_{i-1}$ is less than $\delta$.
Note that since there are only finite number of $x_m$, there is $N\in \mathbb{N}$ such that if $n\geq N$ then $|f_n(x_i)-f(x_i)|<\epsilon$ where $i=0,\cdots, m$
Now if $x\in[0,1]$, then $x\in[x_{i-1},x_i]$ for some $i\in\{1, \cdots m\}$.
My question is how to use the nondecreasingness to arrived at this inequality, for $n\geq N$
$f(x_{i-1})-\epsilon<f_n(x)<f(x_{i-1})+2\epsilon$
Can someone please help, I have been staring at the inequality for about a day now. Thanks.
 A: For the first part of the inequality we know that there is a natural number $N$ such that 
$$ n\geq N \implies |f_n(x_{i-1}) - f(x_{i-1})| < \epsilon$$ 
Opening up the absolutely value signs, this means in particular that for $n \geq N$,
$$-\epsilon + f(x_{i-1}) < f_n(x_{i-1}).$$
Now suppose that $x \in [x_{i-1},x_i]$. Then this means that $x_{i-1} \leq x$ and because each $f_n$ is non-decreasing, for all $f_n$ such that $n$ is sufficiently large we have that 
$$f_n(x_{i-1}) \leq f_n(x)$$
and so using the first inequality I obtained, we have that 
$$f(x_{i-1}) - \epsilon < f_n(x).$$
Edit: Thanks to bobokinks we can now finish the problem. From you inequality we get that there is $N \in \Bbb{N}$ such that for all $n \geq N$, we have
$$|f_n(x) - f(x_{i-1})| < 2\epsilon.$$
I was an idiot before and proved nonsense. Here is the correct version: We have $$\begin{eqnarray*} |f_n(x) - f(x)| &<& |f_n(x) - f(x_{i-1})| + |f(x_{i-1}) - f(x)| \\ 
&<& 2\epsilon + \epsilon\\
&=& 3\epsilon.
\end{eqnarray*}$$
The latter quantity being less than $\epsilon$ comes from uniform continuity of $f$ on $[0,1]$ and the fact that we chose each $[x_{i-1},x_i]$ to be of width less than $\delta$. Since $\epsilon > 0$ was arbitrary, we have proven that for any $x \in [0,1]$, as long as $n \geq N$ we have $f_n \longrightarrow f$ uniformly.
A: BenjaLim has already dealt with the first half of the inequality. For the second half, note
$$f_n(x)-f(x_{i-1})\leq f_n(x_i)-f(x_{i-1})= (f_n(x_i)-f(x_i))+(f(x_i)-f(x_{i-1}))<\epsilon+\epsilon=2\epsilon.$$
