Re: A basic question on real numbers

On 13 Jun 2006 00:04:55 -0700, aone1504@xxxxxxxxx wrote:

I think I have not made myself clear. My question is from completeness
point of view. Rationals(Q) are generated by dividing whole numbers(N).
Irrationals are limits of convergent sequences of rationals. Let's
denote the set of irrationals obatained this way as IR1.

In fact, as several people have pointed out, _every_ real
number is the limit of a sequence of rationals. So your
IR1 is the set of all irrationals.

Now, if we
write a convergent series of irrationals taken from this set IR1 only,
will the resulting value fall within IR1 or Q or N only. Or can it be
outside this set ?

So there are no real numbers outside the union of Q and IR1 - a real
number is either rational or irrational.

In other words, do reals have numbers from only N, Q
and IR1. Is there a proof for this.? I am not very clear whether the
proofs you have given establishes this.

I posted a proof that every real number is the limit of a
sequence of rationals. Was there a step you didn't follow?

Thanks very much.


Dave L. Renfro wrote:
aone1504@xxxxxxxxx wrote:

I have the following basic question:-

Irrational numbers can be obtained as the limit of
cauchy sequence of rational numbers.

Is there a proof that a convergent infinite series
having irrational terms converges to a number which
can always be obtained as the (above) limit of cauchy
sequence of rationls.

I would like to see the proof for this statement.

Let {x_n} converge to x, where each x_n is irrational.
For each n, choose a sequence {r_n_m} of rationals
that converges to x_n. Then the diagonalization of
these sequences of rationals -- namely, the sequence
r_1_1, r_2_2, r_3_3, ... -- is a sequence of rationals
that converges to x. This isn't immediately evident
(to a beginner, at least), but the proof also isn't
very difficult. More generally (same proof method),
a limit of limits is a limit, or in other language,
the (sequential) closure of a set B can be obtained
by (one application of) collecting together all the
limits of convergent sequences in B.

Dave L. Renfro


************************

David C. Ullrich
.

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