I am preparing for my analysis finals tomorrow and I have been stuck over two hours trying to solve the following problem:
Let $f_1, f_2, \ldots$ be a convergent (pointwise) sequence of monotonically increasing functions defined on $[a,b] \to \mathbb R$. (I.e., $f_n(x) \leq f_n(y)$ if $x \leq y$). Let $f$ be the limit of the above mentioned sequence. Assume $f$ is continuous. Show that the above sequence is uniformly convergent.
I am not sure how to approach the above problem. I have been trying to show the above by showing that the window of values taken by $f_N$ (given by $f_n(b) - f_n(a)$) converges and messing around with triangle inequalities to get the required inequality ($f_n(p) - f(p) < \varepsilon$). But this I realized was wrong because even if the windows converge the functions themselves can be increasing at different rates within the interval thus the ($f_n(p) - f(p)$ need not shrink at a constant rate at all points). I feel that the fact $f_n$ is defined in a compact interval and therefore is uniformly continuous comes into the picture somehow, but I can't connect the dots. Any suggestions?