I've been doing a few practice problems, and I've got some that I am stuck on, one of which is (paraphrasing):
Show that if $f_n$ is a sequence of increasing functions on $[0,1]$ that converges point wise to $f$, and $f$ is continuous then $f_n$ converges uniformly.
It look to me like I can use Egorov's theorem and then examine the points which do not converge uniformly, and use continuity from there, but I can't seem to end the problem. I have (for $y$ in the uniform convergence set and $x$ outside it:
$|f_n(x) - f(x) + f(y) - f(y)| \leq |f_n(x) - f(y)| + \epsilon \leq |f_n(x) - f_n(y)| +\epsilon + |f_n(y) - f(y)| $
Where the last term on the right hand side converges uniformly, my problem is if I try and get the first term to converge uniformly then I seem to just go round in circles. Any advise?