Re: abundance of irrationals!)

W. Mueckenheim wrote: 
Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote in message news:<ITSnetNOTcom#virgil-DFC74B.18580709042005@xxxxxxxxxxxxxxxxxxxxxxxx>...


Where did you ever see a definition of oo
SUM (1/2^k) = 1 k=1
which requires one to have k take on any value not a natural number?
The symbol "oo" here simply means "without any upper limit".

In the same way for real limits "as x -> oo" one does not mean that x ever REACHES "oo", merely that there is no upper bound for x. 


So you are convinced that 1/9 is a term the following sequence of partial sums?

0,000...
0,1000...
0,11000...
0,111000...
...

Regards, WM 

I don't think Virgil said anything to imply such a statement. What he appears
to have said is that 1/9 is the limit of a sum of increasing numbers of terms.
There is no need for 1/9 itself to be included in the sequence.

In your discussion of Cantor's diagonal proof, you display the number 0.11111...
This number is the limit of the sequence 0.1,0.11,0.111, ... (that is, the rows
of your list). It is precisely *because* this number 1/9 is not in the sequence
of rows that we say it is not on the list..
.

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