OwlHoot
Posts:
594
Registered:
6/25/06
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Re: Proof 0.999... is not equal to one.
Posted:
Jun 2, 2007 4:12 AM
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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.
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