I'm taking the limit of any convergent infinite sum of terms each weighted by \epsilon. Although \epsilon itself equals zero, of course the limit of the sum is

I'm taking the limit of any convergent infinite sum of terms each weighted by \epsilon. Although \epsilon itself equals zero, of course the limit of the sum isn't necessarily zero. So if we go and define some other funky "infinitesimal" \epsilon != 0 in the surreals, we have to be careful. Apparently people have done that kind of thing successfully, but it took a long time after Cauchy for that to happen.