/sci/ - Science & Math

suppose S = 1+2+... = -1/12
then T = S-1 = 2+3+... = -13/12
and R = T - S = 1+1+... = -1
but R+1 = R = R-1, so
... = -2 = -1 = 0 = 1 = 2 = ...

since the two rationals r/s and p/q are equal if r*q = p*s, we have
|{ p/q : p, q are in N with q != 0 }| = 1

finally, if we construct the real numbers using dedekind cuts, we find that each cut is the same set. hence all real numbers are equal