I detest the algebraic "proofs" used in debates on this.

Anybody who thinks that 0.999... does not equal 1 should probably also reject the idea that 1/3 is the same as 0.333... the argument for both is the identical (getting infinitely closer to some x is equivalent to x itself). You're using the thing you set out to prove as an argument in your proof.

Online debates on this subject would be a lot better if the (correct) side of 0.999... = 1 was more open about the shortcomings of these simpler proofs.