Does 0.9999... = 1 really?

Hey I was talking to this gay guy and he was trying to convince me that 0.9999.... = 1. Is this true?

Whats your take?

.99999999~ = 1 in the set of Real Numbers:

.999999~ means "sum as i goes from 1 to Infinity of 9/10^i". A sum to infinity is defined as the limit of partial sums, and we all agree that the limit is 1.

QED

Another proof:

One of the axioms of the real numbers states that between any 2 real numbers, there are an infinite number of other real numbers. Let S = .99999~ and let T = 1. Assume they were unequal. Then they would have a positive real number as the difference, W. But for any real number you pick, the difference is less than your number(intuitively, just "increase the number of 9s"). The only real number less than all positive numbers is 0.

QED