While it’s easy to prove rigorously I think the easiest way to think about it is to think what number/numbers could go between 0.9 repeating and 1.
I guess that it also requires a second thought experiment that if no numbers can go between two numbers, those two numbers must be the same.

edit: this might be even easier:

1/3 = 0.333… -> multiply both sides by 3
3/3 = 0.999…
1 = 0.999…