Proofs that 0.9999... = 1

Let x = 0.9999...

10x = 9.9999...

10x - x = 9.9999... - 0.9999...

9x = 9

x = 1

QED

Another way:

1/3 + 1/3 + 1/3 = 1

1/3 = 0.3333....

0.3333... + 0.3333... + 0.3333... = 1

0.9999... = 1

QED

One more way:

Let y = 0.1111....

10y = 1.1111...

10y - y = 1.1111... - 0.1111...

9y = 1

y = 1/9 

9y = 9(0.1111....)

9y = 0.9999...

1 = 0.9999...

QED

There are probably some uses for this insight, but I can't think of any off the top of my head.

All repeating decimals can be written as factions, and 1 is the same as the fraction 1/1. 

It must be the case that 0.9999... = 1 otherwise there is a repeating decimal that has no fractional counterpart.