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Topic: Finally the discussion is over: S = Lim S is a bad definition.
Replies: 2   Last Post: Oct 5, 2017 4:42 PM

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Posts: 5,511
Registered: 9/25/16
Re: Finally the discussion is over: S = Lim S is a bad definition.
Posted: Oct 5, 2017 4:42 PM
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Now there is a simple proof, that every Cauchy sequence
is equivalent to at least one decimal representation.

The decimal representation is simply
the limit of the sequence:

bk = floor(a*10^k)*10^(-k)

But the above would be circular, since "a" above is the
real number corresponding to the Cauchy sequence we

want to decimally represent. Can we extract the decimal
representation from a Cauchy sequence without the

detour of a real? Since the sequence converges, we
have a function N(e) such that:

forall n>=N(e) |an-a| < e

When can we use aj instaed of a to compute bk? We
need to assure the following where fk=bk*10^k and
fk is an integer:

fk =< a*10^k < fk

fk =< aj*10^k < fk

Use fk=floor(aj*10^k) and compute the distance:

d = min(aj*10^k-fk,fk+1-aj*10^k)

And check:


Always increment j, but increment k when the
check succeeds. Does it work?

Am Donnerstag, 5. Oktober 2017 21:09:12 UTC+2 schrieb
> Thats a little bit far fetched.
> Decimal representation is a more narrow notion than series.
> Decimal repreentations are only series of the form:
> d0.d1 d2 d3 .... with di in {0,..,9}
> Or if you want you can write it:
> d0 + d1/10 + d2/100 + d3/1000 + ...
> You "decimal" means 10, so this a base 10 digit series.
> On the otherhand this one here is not a decimal, base 10,
> representation:
> 1/2 + 1/4 + 1/8 + ... = 1
> Got it?
> Am Donnerstag, 5. Oktober 2017 20:46:07 UTC+2 schrieb John Gabriel:

> > Also, consider that you don't have a unique decimal representation of any number if you define the series to be the limit. For example, both of the following series have the same limit:
> >
> > 0.999... = 9/10 + 9/100 + ...
> > 0.875... = 1/2 + 1/4 + 1/8 + ...
> >
> > So if we define them both as 1, then 0.999... = 1 and 0.875... = 1. How do we identify what series we are dealing with if we only have the limit? There are innumerably many series with the same limit of 1.
> >
> > On the one hand you berate the limit and on the other hand you invoke it. What is more important, the series or its limit? You seem to think the limit does not matter and then you go right ahead and DEFINE the series to be equal to its limit?! Isn't that stupid?
> >
> > A series is NOT a LIMIT. In fact, no matter if you could hypothetically sum all the terms of 3/10 + 3/100 + ..., you would never arrive at 1/3 because 1/3 is not measurable in base 10.
> >
> > Do you understand that expression "measurable in base 10"? It means expressing any rational number using a given base. There is no fraction p/q such that q=10^n with n integer and p/q = 1/3. It's impossible my little stupid. There is a theorem stating this. You defining S = Lim S goes against the theorem. You just can't use ill-formed definitions. They break everything.
> >
> > Get it?

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