The easy way:
x = .999…
10x = 9.999…
10x - .999… = 9
9x = 9
x = 1
.999… = 1

Other way:

Lets say y is the .99999 number of finite number of 9's (n).
For any y, there is a y1 such that y<y1<1.
We know that y limits to 1 as n goes to infinity, and therefore so does y1 and they're both 1.

If you pick a number thats closer to 1 than my .999999 (n times), I'll simply pick a number closer to 1. We'll do this infinitely until we both end up with 1.

The kicker is that you're doing this infinitely and thats something thats hard to imagine.