.999 ~ 1, Revisited.

I'm sure some of you remember this trending debate on the Internet, once upon a time: As .999... keeps continuing, it eventually equates 1, thus it can be said that .999 ~ 1. However, it is always an infinitesimally .1 decimal point away from becoming one, due to the nature our decimal system. Yet, a fractional ratio proof who's that while 1/3 ~ .333, 2/3 ~ .666, 3/3 !~ .999 . . . being a ratio, 3/3 equates to 1. So now I've brought you up to speed.

Today, I present to you a different scenario. Let's extend the same premise, but instead think of it in a different number base system (i.e., hexadecimal, octal, binary).
Below, is binary.

0.111 ~ 1

Prove me wrong.