In order that that other thread can get locked, here's something for JonF et al (sans organic detritus). Firstly, 0.999... is the same as 9/10 +9/100 +9/1000+.... that is what the decimal expansion means. This sum is equal to 1. Now the other quesition involved taking the limit as n tends to infinity of (0.99...)^n now this is ne, but perhaps, JonF, you're switching limits, which you cannot do without proving it is valid. Meaning here, if x_m = 0.99999...9 with m nines that lim_n ( lim_m (x_m)^n) = 1, yet lim_m (lim _n (x_m)^n) =0 this is ok, and not contradictory if you remember that maths is just the manipulation of symbols to follow certain rules. One is that you cannot exchange limits whenever you feel like it. There are times you can times you can't. Here's a couple of examples I gave in another thread today where it matters when you take limits. for the interval [0,1] define f_n to be x^n the point wise limit of this os the function f where f(x) = lim_n f_n(x) which is the same as the function that is 0 for all x in[0,1) and 1 at 1. So limits don't commute with continuity. the second is g_n defined on R as 1+x/n for x in[-n,0] 1-x/n for x in[0,n] 0 elsewhere. the pointwise limit is the zero function and the integral of that over R is zero, but the integral of each g_n is 1, so limits don't commute with integration.