0.999...=1 ?

Perhaps not perfect proof, but I would do as follows:

x = 0.9999... |multiply both sides by 10

10x = 9.9999...

Now 10x - x = 9.9999... - 0.9999..., so

9x = 9,

Thus x = 1

This is not a formal proof, it is more of a demonstrations

You can show by division that                        $${\frac{{\mathtt{1}}}{{\mathtt{3}}}}$$    = 0.3  repeater.

If you multiply both sides by 3 you get            $${\frac{{\mathtt{3}}}{{\mathtt{3}}}}$$    = 0.9 repeater

but      $${\frac{{\mathtt{3}}}{{\mathtt{3}}}}$$   = 1       so

                                                                 1 = 0.9 repeater