/sci/ - Science & Math

>>15140972
Okay.
I imagine you're most of the way done with the exhaustion proof by now and feel that I should mention that you can write down [math]\displaystyle f_n(x) = \int_0^x n \chi_{[0, 1/n]} (t) \ dt[/math] and prove from there that [math]\displaystyle |f_n (x) - f_n (y)| = \left| \int_0^x n \chi_{[0, 1/n]} (t) \ dt - \int_0^y n \chi_{[0, 1/n]} (t) \ dt \right| = \left| \int_y^x n \chi_{[0, 1/n]} (t) \ dt \right| \leq n |x - y|[/math]
The general result, as you've probably noticed, is that integrals of bounded functions are Lipschitz.