Who is asking: Student
While there are various arguments that 'show' this equality is the 'right way to think about it', my favourite is the following.
IF the two numbers 0.99999... and 1 were NOT equal, you would have a number to represent the difference (the GAP or Distance between them) - and it should not be 0.
So what is the difference?
It is a number smaller than 0.0001, smaller than 0.000001, ....
So this really comes out to the question: is there a number smaller than 0.0001, 0.0000001, .... but bigger than 0?
When you check there is no such number. The gap between them really is 0. Of course it is a bit obscure since we are dealing with 'infinite processes' (sometimes called limits) whenever we work with decimal numbers which do no terminate. Hard to think about an infinite process with a finite mind and a finite amount of time. However, amazingly, we do manage to do it.