In Math to prove something fully, you also have to disprove it is false. You can not disprove that 0.999999... =/= 1, therefore the statement is not true. (when you disprove falsehood you use indirect proofs usually)

Infinity can never be reached, only approached, and also does not actually exist in math. The full proven statement here is that as the number of digits after the decimal place approaches infinity, the value of the number (which is still rational) approaches 1, but never quite reaches it.