>>8797338

First of all, 0.999... is a rational number since it only consist of a repeating decimal. Thus, it must have a quotient to represent it. Since it can't be directly represented by a quotien, it must be equal to something that can be represented as a quotient.

It can be represented as the infinite sum below.

>9*(1/10)+9*(1/10)^2+9*(1/10)^3...¨

This is the geometric series seen below with a=9 and r=1/10.

>ar+ar^2+ar^3+ar^4+ar^5...

The solution to this geometric series is seen below, when the summation is done to infinity and |r|<1

>ar/(1-r)

We insert the given numbers for the representation of 0.999... and get that it is equal to one.

>(9*(1/10))/(1-(1/10))=(9/10)/(9/10)=1

Even though limits have been used, it can be argued that 0.999...=1 because it has to be equal to something that can be represented as a quotient, because it is by definition a rational number and can't be represented as a quotient in and of itself.