[Burke]

That is actually a unanimously accepted proof, it is just a fact. I think you might not believe at first sight because you are under the notion that any given number has only one decimal expansion, which is not the case. It is as easy to prove in base-3 as it is in base-10. Defining pi would have nothing to do with this, as it only shows that pi is defined and you can represent it with decimal notation. I could define pi as a continued fraction if I like or from bessel functions like Ramajan did: these represent the same number, but will likely have differing based off of convergence.

As a challenge, if .999... is not equal to 1, what is the difference of the two of them (i.e., what is 1-.999...). Recognizing what

lim 1/(10^N) is as N approaches infinity should help you in this endeavor.