You challenged me to demonstrate that mathematical induction is a valid deductive principle…
No. I challenged you to use it to prove 1/2 + 1/4 …= 1. I know about math induction, have used it many times.
…please explain where this proof goes wrong:
It goes wrong in the very first line, because two of the symbols are undefined [i.e. 0.9999… and 9.9999…], unless you go with all mathematicians everywhere and define them as least upper bounds. Then the whole thing becomes a proof about least upper bounds, with the final result being that the l.u.b. of .9999… is 1.
But to say that a convergent infinite sum that has not been decomposed or rearranged in any way is not identically equal to its numerical summand is ridiculous.
I agree, with the proviso [that you deny] that a “convergent infinite sum” means, by definition, a least upper bound. But no more than that. Corrolary: The sqaure 1x1 has not been proven anywhere that it will be filled by dropping in squares of 1/2 , 1/4, etc.
No serious mathematician except perhaps a few finitists would ever say such a thing.
So Hardy wasn’t a serious mathematician. And every mathematician on the subject since the 19th century is also not serious. Clayton, time to put up or shut up. Show me one, just one, book, website, link, something, that says what you do.
Contrary to popular belief, analysis does not asterisk the equality between the infinite sum and its summand, it simply analyzes the behavior of the partial sums.
Contrary to popular belief [as exhibited in this thread], analysis does not define an infinite sum in any other way but as a least upper bound. There is no other infinite sum. The infinite sum is distinct from its summand, because the infinite sum is the limit of the summand [which is an infinite sequence].
The mistake lies in an ignorance of the meaning of a summation sign that has an infinite index. Whenever you see one, it means only one thing, by definition. It means the limit of a certain sequence.
And I did not misquote Hardy - I intentionally inserted the “also”…
Oh. OK. You mean you did not realize that inserting that innocent looking word negates everything he actually wrote.
It is you, not me, who is misreading him.
OK then. I leave it to whoever is reading this to go straight to the master [=Hardy], sit at his feet [=read Chapter 4 and beginning of Chapter 8] and see what he says.
We are done here, Clayton. If we disagree on the meaning of what Hardy actually said, there is no room for further discussion.