0.999... = 1?

My stupid math prof tells me that .9999999... = 1. But I think he is just stupid.

So, what's your take?

0.999... = 0 + 9/10 + 9/10^2 + 9/10^3 + ... = (9/10) * ( 1 + 1/10 + 1/10^2 + 1/10^3 + ... )

This is obviously the geometric series, so we get:

(9/10) * ( 1 + 1/10 + ... ) = (9/10) * ( 1 - 1/10^n ) / (1 - 1/10 ), for n -> inf.

Since 1/10^n -> 0, we get:

(9/10) * ( 1 - 1/10^n ) / ( 1 - 1/10 ) = (9/10) * 1 / ( 1 - 1/10 ) = 9/10( 1 - 1/10 ) = 9 / ( 10 - 1 ) = 9 / 9 = 1

q.e.d.