You are right. The argument is that it can approximate sin(x) over a compact interval, like [0,1]. You answer to that might be "I could approximate over [0,1]

You are right. The argument is that it can approximate sin(x) over a compact interval, like [0,1]. You answer to that might be "I could approximate over [0,1] with just a lookup table, where I split the input range into n equal sized pieces for increasing values of n", and you'd be right -- the "proof" is basically just doing that. It's one of those things which is nice to show as a basic theory thing (there are approximation methods which can never simulate certain functions), but it's not really of any real value.