1.3 The Real Numbers

Since the concept of continuity with respect to the Real Numbers is so central to what follows, a brief review/introduction to the construction of the Real Numbers from the Rationals is in order.

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The Definition of the Real Numbers as Dedekind Cut:

A Dedekind Cut is a partition of the Rational Numbers into two non-empty sets $\QTR{Large}{r}$ and MATH ($\QTR{Large}{r}$ $\QTR{Large}{\cup }$ MATHand $\QTR{Large}{r}$ $\QTR{Large}{\cap }$ MATH) such that:

  1. If MATH and MATH then MATH .

    ( Hence, if MATH and MATH then MATH)

  2. For every MATH there exists a MATH $\QTR{Large}{\in r}$ such that MATH

Notes:

Property 2. allows us to avoid the usual ambiguities caused by the "fact" that MATH and MATH

are the "same" real number.

In general in what follows, rather than explicitly computing MATH as the complement of $\QTR{Large}{r}$ , when appropriate we will restrict our definition of a cut to the definition of $\QTR{Large}{r}$.

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Some Basic Properties of Dedekind Cuts:

  1. There is a natural inclusion of the Rationals in the Reals.

    MATH

    Note, that MATH

     

  2. The Real Numbers are ordered. (See the example below)

    MATH if the exists a MATH such that for all MATH , MATH

    Or equivalently, MATH as sets

    Or equivalently, MATH as sets

     

  3. One can extend the operation of addition from the Rational Numbers to the Reals.

    MATH and MATH $\QTR{Large}{\}}$

     

  4. One can extend the operation of negation from the Rational Numbers to the Reals.

    For $\QTR{Large}{q}$ a Rational, define MATH. If MATH for any $\QTR{Large}{q}$ , define MATH $\QTR{Large}{\}}$

     

  5. Every set of Real Numbers bounded above has a least upper bound.

    One verifies that, in general, MATH satisfies property 1. and 2. of the definition of a Dedekind cut.This follows from

    the fact that the properties hold for each MATH.

    Now, let MATH be a set of Reals and suppose that for all MATH.

    We need to check that MATH is not empty. But, again, it is a simple argument to show that MATH

     

A Dedekind Cut is a partition of the Rational Numbers into two non-empty sets $\QTR{Large}{r}$ and MATH ($\QTR{Large}{r}$ $\QTR{Large}{\cup }$ MATHand $\QTR{Large}{r}$ $\QTR{Large}{\cap }$ MATH) such that:

To be turned in Tuesday January 34: Complete the proof of 5.

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First we need to show that

  1. If MATH and MATH then MATH .

    ( Hence, if MATH and MATH then MATH)

  2. For every MATH there exists a MATH $\QTR{Large}{\in r}$ such that MATH

1. From Boolean Algebra we know that MATH

So if MATH MATH for all MATH. Hence for all MATH and all MATH in particular for MATH , we have. MATH .

2. If MATH then for some MATH and all MATH There there exists a MATH MATH such that MATH

But then MATH

What is left to show is that MATH. But MATH for all MATH MATH.

Finally we show that is a least upper bound.

That it is an upper bound is amounts to observing that MATH $\QTR{Large}{=r.}$

That is a least upper bound follows is also immediate since MATH for all MATH for all MATH MATH

MATH

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Example:

To show MATH MATH we need to find some MATHsuch that MATH . In particular we need to find $\QTR{Large}{q\ }$such that MATH .

Let MATH