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One way to tell if two numbers are equal is to show there’s no real number between them. Try to formulate a number that’s between 0.999… and 1. You can’t do that.
But between 0.999 and 1 is 0.9999.
If something comes ever increasingly close to, but never physically touches something else, would you say it’s touching it?
0.999… means infinitely repeating 9s. There’s no more 9 to add that hasn’t already been added. If you can add another 9, then it’s not infinitely repeating.
So it never ends, and it stays 0.9… infinitely?
Still not a 1.
It's an infinite number of nines after the decimal.
Or think of it another way. What number would you subtract from 1 to get 0.999... ? The answer is 0.
let x = 0.999...
so 10x = 9.999...
subtract first line from second:
9x = 9
divide by 9
x = 1
an asymptote 😎