3
$\begingroup$

I have the following theorem to prove.

Given a real number $a$, define the set $S$ such that $S = \{x \in \mathbb Q: x < a\}$. Show that $a = \sup S$.

My attempt at a proof is as follows

Proof: Let $m$ be a rational number larger than $a$. Then by the definition of the set, this element is excluded from $S$. As this element is larger than every element in $S$, this is an upper bound for $S$. Therefore $S$ has at least one upper bound. As $S$ is bounded, by the Completeness Axiom, it has a least upper bound and therefore has a supremum.

!! This is where I get stuck, I know intuitively that the number a must be the least upper bound, but I'm not exactly sure why. I know that there must be some property of the reals that is responsible for this almost obvious fact but I'm not sure what it is.

I have a feeling that this is so simple that you might accidentally answer it for me in the process of helping, but if you could offer a hint that might lead this horse to water without making it drink I would be greatly appreciative.

$\endgroup$
1
$\begingroup$

What you have done so far is fine. $S$ is bounded above, so it has a supremum. Let's call that supremum $M$. We need to show that $M=a$. To do this, we can rule out the other two possibilities, namely $M<a$ or $M > a$.

If $M < a$, then can you find an element $x\in S$ with $M < x < a$? If so, then $M$ isn't even an upper bound, let alone the supremum.

If $M > a$, then since $M$ is the least upper bound of $S$, there must be some $x\in S$ with $a < x < M$. What can you conclude?

| cite | improve this answer | |
$\endgroup$
0
$\begingroup$

I would suggest a proof by contradiction. Suppose that there exists a real number $b \in \mathbb{R}$ such that $b$ is an upper bound of $\{ x \in \mathbb{Q} : x < a \}$. How do we know there exists a rational number $c$ such that $b < c < a$?

| cite | improve this answer | |
$\endgroup$
0
$\begingroup$

'... it is of interest to note that on 9 February 1918 Weyl and George Polya made a bet in Zurich in the presence of twelve witnesses (all of whom were mathematicians) that "within 20 years, Polya, or a majority of leading mathematicians, will come to recognize the falsity of the least upper bound property." When the bet was eventually called, everyone - with the single exception of Godel - agreed that Polya had won.' The Continuous and the Infinitesimal, John L Bell, p225.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Nice anecdote! :-) $\endgroup$ – Bungo Sep 12 '14 at 5:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.