Well, I'm not a big fan of this "infinite magnification" idea in the first place, but "arbitrarily close" is typically one of those things that infinity can beat.

(Compare e.g. with the fourier transform of a function. It consists of a sum series which comes "arbitrarily close" to the function, but "at the limit" when the number of terms approaches infinity the function and its fourier transform is one and the same.)