Proof that .9999... = 1



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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