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Math Help - Least Upper Bound Property

  1. #1
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    Least Upper Bound Property

    In most books, the LUB property of \mathbb{R} is treated as an axiom and then it is shown that it is equivalent to completeness. However, in baby Rudin, it appears that in defining \mathbb{R} as the set of Dedekind cuts, we prove that \mathbb{R} has the LUB property directly from the definition.

    Likewise, in defining \mathbb{R} as the set of equivalence classes of Cauchy sequences (under the relation that two sequences are equivalent if and only if the sequence of differences is null), we show directly from the definition that \mathbb{R} is complete.

    Here is what confuses me: Why is it considered an axiom if we can prove each of these (equivalent) properties from the equivalent definitions of \mathbb{R}?
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  2. #2
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    Quote Originally Posted by patrick View Post
    In most books, the LUB property of \mathbb{R} is treated as an axiom and then it is shown that it is equivalent to completeness. However, in baby Rudin, it appears that in defining \mathbb{R} as the set of Dedekind cuts, we prove that \mathbb{R} has the LUB property directly from the definition.

    Likewise, in defining \mathbb{R} as the set of equivalence classes of Cauchy sequences (under the relation that two sequences are equivalent if and only if the sequence of differences is null), we show directly from the definition that \mathbb{R} is complete.

    Here is what confuses me: Why is it considered an axiom if we can prove each of these (equivalent) properties from the equivalent definitions of \mathbb{R}?
    I believe what you're saying is that why, although not taken to be an axiom, is the LUB in Rudin considered one? I think the answer lies in the fact that \mathbb{R}\text{ is complete }\Leftrightarrw \mathbb{R}\text{ has the LUB property}. Thus they are logically equivalent and interchangeable in sense.
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  3. #3
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    What is, or is not, an axiom depends on where you are starting. If you are defining the real numbers as a field itself, axiomatically, with no prior information, then you must take the "least upper bound" property as an axiom- it "defines" the real numbers

    But you can also define the rational numbers first, axiomatically, then construct the real numbers system from the rational numbers. That is what Rudin does, using Dedekind cuts. That way, you can prove the "least upper bound property" from the definition of Dedeking cuts and the properties of the rational numbers.

    You can also construct the real numbers using Cauchy sequences- say that two Cauchy sequences, \{a_n\} and \{b_n\} are "equivalent" if and only if the sequence \{a_n- b_n\} converges to 0 and then define the real numbers to be equivalence classes of such sequences. That makes it easy to prove the "Cauchy Criterion", that all, in the real numbers, all Cauchy sequences converge, which is equivalent to the least upper bound property (in the sense that given either one, you can prove the other).

    A third way to construct the real numbers is to use "increasing sequences with upper bounds". We define "equivalence" for two such sequences in the same way and define the real numbers to be the equivalence classes. That makes it easy to prove "monotone convergence" which is equivalent to both "Cauchy Criterion" and "least upper bound property".
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    Quote Originally Posted by HallsofIvy View Post
    What is, or is not, an axiom depends on where you are starting. If you are defining the real numbers as a field itself, axiomatically, with no prior information, then you must take the "least upper bound" property as an axiom- it "defines" the real numbers

    But you can also define the rational numbers first, axiomatically, then construct the real numbers system from the rational numbers. That is what Rudin does, using Dedekind cuts. That way, you can prove the "least upper bound property" from the definition of Dedeking cuts and the properties of the rational numbers.

    You can also construct the real numbers using Cauchy sequences- say that two Cauchy sequences, \{a_n\} and \{b_n\} are "equivalent" if and only if the sequence \{a_n- b_n\} converges to 0 and then define the real numbers to be equivalence classes of such sequences. That makes it easy to prove the "Cauchy Criterion", that all, in the real numbers, all Cauchy sequences converge, which is equivalent to the least upper bound property (in the sense that given either one, you can prove the other).

    A third way to construct the real numbers is to use "increasing sequences with upper bounds". We define "equivalence" for two such sequences in the same way and define the real numbers to be the equivalence classes. That makes it easy to prove "monotone convergence" which is equivalent to both "Cauchy Criterion" and "least upper bound property".
    Thanks for replying. After thinking about the "equivalence classes of Cauchy sequences" definition, it seems difficult to actually make this into a metric space without using completeness. For instance, we need to define the distance between to equivalence classes of Cauchy sequences: The natural way to do this is by setting

    <br />
d_{\mathbb{R}}\left(\left[x_{n}\right],\left[y_{n}\right]\right) = \lim_{n\rightarrow\infty} d_{\mathbb{Q}}\left(x_{n},y_{n}\right) .<br />

    Now, we can easily show that \left\{d_{\mathbb{Q}}\left(x_{n},y_{n}\right)\righ  t\} is a Cauchy sequence, but we don't know that the limit actually exists without assuming that \mathbb{R} is complete (which unless taken as an axiom we do not know). Therefore, on second thought, I don't see how one can show that with \mathbb{R} defined in this way, it is complete (ie. every Cauchy sequence converges).

    If we know that the metric I wrote above exists, then I agree that we can show that \mathbb{R} is complete. However, we don't know that such a metric exists without knowing \mathbb{R} is complete, so it seems completely hopeless to define the real numbers in this way. Do you have any other ideas on how to define a metric and show that the space is complete as you suggested?
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