The Wikipedia article says how, you need a definition of real numbers that includes nonzero infinitesimals (IOW does not satisfy the Archimedian property).

So let there be an ω with 0.999... = 1 - 1/ω. Then a number between 0.999... and 1 would be 1 - 0.5/ω.

How do you know that 1/ω is not 0?

{ 0.999... | 1 } using surreal number notation.

? Just because there is nothing between two numbers does not mean the two numbers are equivalent. What nonsense is this

Either two numbers are equal, or they're not equal. (Unless we're calling into question the law of excluded middle, but must we?)

Without using 9999..., please name any two real numbers that don't have a number in between.

That is true for the reals