A uniformly convergent sequence of continuous functions converges to a continuous function. I have no problem with the conventional proof. However, in Henle&Kleinberg's Infinitesimal Calculus, p. 123 (Dover edition), they give a nonstandard proof, and they use the hyperhyperreals to do it. I don't understand why that's needed. The proof is very short; here's an outline: Let '~' mean infinitesimally close to and '~~' mean hyperinfinitesimally close to (i.e., infinitesimal as viewed from within the hyperreals, using the hyperhyperreal system constructed on top of the hyperreals). Given a functions fn → (uniformly) f, and given any real x, THEN for any h ~~ x, and for any infinite N, we'll have: fN(x) ~~ fN(h) (by assumption), fN(x) ~ f(x) (by assumption), and fN(h) ~ f(h) (by assumption). Then we have f(x) ~ fN(x) ~~ fN(h) ~ f(h), and so f(x) ~ f(h), and f is continuous at x. I don't understand why the proof doesn't work equally well if we just use the hyperreals, and replace the '~~' with '~' everywhere. Any clue would be appreciated.