The proof follows from the definition I gave for equality, where we look at the difference between the two numbers. In this case
we want the difference between 1 and 0.9999recurring. Now, that recurring is a bit tricky, so let's take a series of finite approximations

1-.9 = 0.1
1-.99 = 0.01
1-.999 = 0.001
1-.9999 = 0.0001

You can see where this is going, I hope. For any epsilon you care to pick, let's say, 0.00000000000000000000000000000000001, or one with a hundred million leading zeros before the '1', I can produce a term in this series where the difference is smaller, because the 'recurring' means I can write as many nines as I like, even if it is a hundred million and one. And since that's my definition of equality, the two representations denote the same number.