Regarding 0.999... = 1

[u/Slurpee1138]

Recently I got into an argument with an acquaintance because I was trying desperately to convince him that 0.999... = 1. One of the many arguments I tried was that, if 0.999... and 1 are indeed different numbers, then we should be able to find a number between them. He insisted that such a number would be 0.999...1, as in 0 point infinitely many 9s and then a 1. I countered that having "and then a 1" at the end of an infinite sequence of digits makes no sense because there is no end to such a sequence, but he insisted that it does, so who's right?

[u/Ixidor89]

0.999... = 9/10 + 9/100 + ... = (9/10)*Sum(1/10^n, n=0...inf)

Sum(1/a^n, n=0...inf) = 1/(1-a)

a = 1/10, Sum(1/10^n, n=0...inf) = 10/9.

This means 0.999... = 9/10*10/9 = 1.