Earlier quoted context omitted.
I like the argument that observes "if you subtract 0.99(9) from 1, you get a number in which every decimal place is zero". The geometric series proof is less fun but more straightforward. As a fun side note, the geometric series proof will also tell you that the sum of every nonnegative power of 2 works out to -1, and this is in fact how we represent -1 in computers.
How can the sum of a bunch of positive powers powers of 2 be a negative number? Isn't the sum of any infinite series of positive numbers infinity?
\2 is "not always" ..
Consider SumOf 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ...
an infinite sequence of continuously decreasing numbers, the more you add the smaller the quantity added becomes.
It appears to approach but never reach some finite limit.
Unless, of course, by "Number" you mean "whole integer" | counting number, etc.
It's important to nail down those definitions.