OwlHoot
Posts:
594
Registered:
6/25/06


Re: Proof 0.999... is not equal to one.
Posted:
Jun 2, 2007 4:12 AM


On May 31, 7:16 am, chaja...@mail.com wrote: > > I have written a proof that 0.999... cannot be equal to one in > the system of real numbers. > > While at the end of it all you may not fully agree with my proof
<groan> Here we go again. ISTR the very first sci.math post, back in c. 1980, was about this, and they'll probably still be discussing it when the last star falls from the Heavens ..
I'm sure people get hung up on the notation '0.999..', thinking of this as a decimal representation and thus one of the elements of the sequence {0.9, 0.99, ..}.
Although technically any decimal has an infinite number of digits, possibly all zero from some point onwards or all '9', it is easier in this context to think of it provisionally as no more than a character string denoting the limit of the above sequence, in the sense one could just as well denote this limit (if it exists) by the word 'sod'.
Forget all that rubbish about subtracting ten times one number from the next and so forth. The only question you need ask is whether you can choose any value, however close to zero, and find that the difference between 'sod' and each number of the sequence from some point onwards is less (in magnitude) than that value you chose.
If this doesn't work for any value, which it won't for example with the sequence {1, 10, 100, ..}, then that sequence has no limit.
If it doesn't work for some value such as 2 with a sequence such as {1, 1/2, 1/3, ..} then you can conclude that 2 is not the limit of that sequence.
If it does work, as it does with { 0.9, 0.99, ..} using the value 1, then that value is our limit. So in short the limit 'sod' equals 1.
Note that none of the numbers 0.9, 0.99, .. equals 1, any more than the limit zero of the sequence 1, 1/2, 1/3, .. is contained in that sequence.
The limit of a sequence need not be any of the numbers in the sequence, although it can be, for example 1 is present (lots of times) in {1, 1  1/2, 1, 1 + 1/3, 1, 1  1/4, ..} whose limit is 1.
Cheers
John R Ramsden
P.S. It is more conventional to denote a limit by a letter such as X, rather than 'sod' or 'bollocks'. I just chose that in a perhaps forlorn attempt to make the point more forcefully.

