Can you prove 0.999... = 1 because 0.999... * 0.999... = 0.999...?

If you were to use just algebra there are only a few times in which x² = x, namely (edit)[0, and 1].

If I calculate 0.999 * 0.999 = 0.998001. (for every 9 you include in the multipliers, there will be x-1 nines in the solution, followed by one 8, then x-1 0s, and finally, a 1.

I'm not at the level of math where I deal with proofs, but I'm pretty sure I can assume that I'm correct in saying: In the equation y = x^2, as x approaches 1 from the left, y approaches 1. So (0.999...)² = 1 and 1² = 1, thus (0.999...)² = 1^2, and finally, ±0.999...= ±1.

Side note: are the ±s needed?