That's a good one! Might replace my current favorite which is:

Let x = 0.99...

Then 10*x = 9.99...

And if we subtract x from both sides, we get:

10x - x = 9.99... - x

And since we already defined x=0.99... when we subtract it from 9.99..., we get

9x = 9

So we can finally divide both sides by 9:

x = 1