Sort of. The error at a square wave's "cliff" does shrink in width as you add higher and higher frequencies to the approximation, but the height of the error do

Sort of. The error at a square wave's "cliff" does shrink in width as you add higher and higher frequencies to the approximation, but the height of the error does not diminish, so you have to be explicit about what you mean by "converge". If you considered MAX(ABS(approximation(x) - square(x))) as your error metric it would not go to zero even in the limit; for most intents and purposes we don't care about that, though, and in any case the limiting behavior is much nicer for "normal" functions that don't have discontinuities like a square wave.