People who offer "proofs" otherwise don't understand math.
That's a harsh comment, but I think it's fair. Consider this common "proof" the two equal one another.
10x - x = 9x
.999 = 1.
The hand-waving is obvious. How does one multiply an infinite series of 9s by 10? What happens to the zero you'd get when multiplying by ten? Are we to believe it just disappears?
Of course not. The proof is invalid. It's just an optical illusion relying on tricking the reader by hoping they don't notice the hand-waving. The reality is no "proof" can address the issue better than simply looking at the two values. If the two are equal to one another, subtracting one from the other must give an answer of 0.
When we do that, we see there is a difference - 0.000...1. It's infinitely small, but it is real. Therefore, the two numbers aren't equal. Why then do so many people believe they are? Because they say so.
Literally. They say so, so it's true. That's it. You see, there is a thing in math called an axiom. It's a statement assumed to be true without proof. One axiom underlying the real number system basically says:
Non-zero infinitesimals do not exist.
Which means 0.000...1 does not exist. Why? Because we say so.
Only we don't say so. A person who thinks 0.999... doesn't equal 1 obviously believes infinitesimals exist. They don't accept that axiom. They'd use a different one, like many mathematicians who work with infinitesimals on a regular basis.
That's right. There's an entire field of math which uses infinitesimals. It's just as valid as the real number system. Which one you use is merely a matter of preference. Whether 0.999... and 1 are equal is based on the completely arbitrary choice of whether one uses the real number system or a different one.
All the "proofs" the two are equal are meaningless. Implicit in all of them is the statement, "Using the real number system." That's begging the question. It's tricking people by assuming there could be no difference between the two numbers then concluding there is no difference between the two numbers.
Anyone who understands how math works should know 0.999... equals 1 only if you choose for it to. "Proofs" the two are equal tell us nothing about the subject but everything about the speaker. Namely, they don't know what they're talking about.
If you feel 0.999... does not equal 1, you're right. If you feel it does equal one, you're right too. Which answer is "right" just depends on which type of math you feel most comfortable with. It's purely a matter of personal choice.