It's not though. There are far better explanations in this thread - explanations that aren't so easily disproven.

0.5 - 0.4999... != 0.0000...

rather

0.5 - 0.4999... = 0.0000...1

Saying

0.0000...0 = 0.0000...1

is an approximation, one you could have just made by saying

0.5 = 0.4999...

The point of the assertion in the OP is that 1 is provably equal to 0.9999..., not that it is approximatelyequal. Proofs require the use of accepted truths to logically arrive at the conclusion. So...

Our accepted truths:

0.9999... = 0.3333... + 0.6666...

0.3333... = 1/3

0.6666... = 2/3

Some substitution:

0.9999... = 1/3 + 2/3

Thus:

0.9999... = 1

For 0.5 = 0.4999..., we can just subtract 0.5 from both sides in the above.

(I can't take credit for this proof because I only learned it today in this thread.)