0.999.../Proof by equality of Dedekind cuts
In the Dedekind cut approach, the real number 1 is the set of all rational numbers that are less than 1. Meanwhile, the real number 0.999... is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the form
Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
Since 0.999... and 1 contain the same rational numbers, they are the same set: 0.999... = 1.