Just want to add that this way of looking at it is only helpful if reals are the agreed upon number set.
If we talk about rational numbers the 1/3 + 1/3 + 1/3 proof is not so bad, is it? Rational numbers can be expressed using repeated decimal notation.
It's not a proof at all, as the author acknowledges. In fact it essentially assumes the consequent.
The issue is that writing down 0.333... essentially already assumes that you are working with real numbers. 0.333... should be thought of as an infinite series and you really want to use analysis in that case.
You can do analysis on the rationals, but that is a very bad idea. But in the end you would still define equality the same way, which already is the proof.