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Let $(f_n)$ be a sequence of monotonically increasing (ie. nondecreasing) functions defined on $\mathbb{R}$ with $0\leq f_n\leq 1$ for all $n$. In addition, $f_n\to f$ pointwise for some continuous $f$.

Baby Rudin (Principles of Mathematical Analysis 3rd Ed.) 7.13.b asks us to prove that the convergence must be uniform, but I think I have a counterexample: put $I(x)=0$ on $x<0$, $I(x)=1$ on $x\geq0$ and $f_n(x)=I(x-n)$ so that $f_n\to0$ pointwise but not uniformly.

Under what extra conditions is the original statement true? For example, require that the bounds $0\leq f\leq1$ be tight.

Edit: clarified definition of monotonic (this is the definition that Rudin uses).

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    In the first edition, the problem is to show uniform convergence on compact sets.2012-12-18

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