[constantcrying]

All these supposed proofs are totally wrong. Students are correctly interpreting as hand waving, by people who themselves do not have a good answer because that is exactly the case.

The reason 0.999... and 1 are equal comes down to the definition of equality for real numbers. The informal formulation would be that two real numbers are equal if and only if their difference in magnitude is smaller than every rational number.

(Formally two real numbers are equal iff they belong to the same equivalence class of cauchy series, where two series are in the same equivalence class iff their element wise difference is smaller than every rational number)