Proof that .9999... = 1

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Let x = .9999... - 1.
If we can show that x = 0, we are done.

Computation

x = .9999... - 1	=	.9999... - .9 + .9 - 1
	=	.0999... - .1
	=	(1/10)*.99999... - (1/10)*1
	=	(1/10)*(.9999... - 1)
	=	(1/10)*x

And since x= (1/10)*x, therefore x=0.

Additional remark: Subtracting .9 from .999... gives

   9999...

- .9         

  .09999... = (1/10)*.9999...

Comment: The equality .09999... = (1/10)*.9999... states that the difference between .9999... and 1 cannot be an infinitesimal.

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