Date: Sat, 23 May 1998 08:22:48 -0400
From: Alex Bogomolny
Dear George:
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.
Best regards,
Alexander Bogomolny
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