It's a pretty easy proof. I'm on mobile so I can't elaborate a lot, but just look how do you construct the fraction corresponding to a periodic number. In this case, it's 9 divided by 9. That's 1.
Another proof (requires more knowledge) is based on the structure of the real numbers. Let a and b be real numbers where a ≠ b. Then, there are infinite numbers between them. It's a pretty obvious statement.
Now, try to find any number between 0.99... and 1. You can't. So, if there isn't any number between 0.99 and 1, they are equal.
Another proof (requires more knowledge) is based on the structure of the real numbers. Let a and b be real numbers where a ≠ b. Then, there are infinite numbers between them. It's a pretty obvious statement.
Now, try to find any number between 0.99... and 1. You can't. So, if there isn't any number between 0.99 and 1, they are equal.