There is no problem with the proof except the assumption that the value in the limit is the same as the value at infinity. If we simply define pi(n) as a function from N U {inf}, which gives the value that "pi" takes at the nth step of the process, and pi(inf) as the value that it actually takes for the circle, then we simply have a function where lim n->inf pi(n) ≠ pi(lim n->inf). For all finite n, it equals 4, and then at infinity it equals 3.1415... .

There are ways to reformulate the above so that "infinity" isn't involved but this is the clearest way to think of it. It isn't much different than the Kronecker delta function delta(t), which is 1 at t=0 and 0 elsewhere. We have lim t->0 delta(t) ≠ delta(lim t->0 t).