I think it's because it looks like you didn't read the article, which explains repeatedly that this is a theorem of universality. Any continuous function will work, and sin is continuous.
Try changing f(x) to "sin(x)" or whatever you like (e.g. sin(10x) may be more interesting). Then drag the sliders for w (how step-like the purple pieces are) and n (how many pieces to add together). The function f(x) is red, and the approximation is blue.
