Subject: Re: .9999... = 1?
Date: Sat, 23 May 1998 08:22:48 -0400
From: Alex Bogomolny
To answer your questions: yes, 9,9999999999...... = 10. Why? From the definition of limit. The infinite fraction on the left makes only sense as a shorthand for that limit which happens to be 10.
You can also write it as 9 + 0,99999... where the second summand is again a limit which is now equals 1. The limit at hand is the limit (usually called the sum) of a convergent series. Covergent series can be multiplied termwise.
For example, 10*9,999... = 99,999... = 99 + 0,999... = 100.
Note, that the termwise multiplication only makes sense after the convergence of the series has been established. But once it has, you automatically know the limit as a sum of a geometric series. Your "proof" may also be used in finding that limit. However, it does not prove that the limit exists and, for this purpose, is not valid.