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Let A and B be subset of the rational numbers defined as follows:

A = {p in Q: either p<=0 or p^2 <2} B = {q in Q: q>0 and q>0 and q^2 >2}

  1. Show that the set B has no smallest element.
  2. Show that the set A fails to have a least upper bound property

My attempt at question 1 is that we try to find an element x in Q(rationals) such that x has form x= q - delta, where q is in B and delta > 0 and q - delta < q. I know that x^2 = (q - delta)^2 > 2.

This is the point where I get stuck. I'm trying to define delta in terms of q and then expanding the term (q-delta)^2 to show that it is greater than 2. I am not too sure if this is the correct and most efficient way to do prove that B has no smallest element.

Help and advice is very much appreciated. Thank you

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It would certainly work. Show that $q^2-2\delta-\delta^2>2$ iff $\delta(2-\delta)<q^2-2$. Since $0<q^2-2$ and $\delta(2-\delta)\mathop{\to}_{\delta\to 0}0$, one such rational $\delta>0$ exists. –  Jonathan Y. Sep 17 '13 at 0:08
    
Thanks for the quick response. I'm having a hard time trying to find an explicit equation of delta in terms of q that will make this inequality true. Any tips to find such a delta? –  tamefoxes Sep 17 '13 at 0:16
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Well, if we accept that $\delta<2$ than one has $\delta(2-\delta)<2\delta$, so it would suffice that $\delta<\frac{q^2-2}{2}$. One such $\delta$ is $\min\{1,\frac{q^2-2}{4}\}$. –  Jonathan Y. Sep 17 '13 at 0:19
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Ahhhh, that makes perfect sense. Thank you very much for all the help you've provided Johnathan. –  tamefoxes Sep 17 '13 at 0:24
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1 Answer

up vote 1 down vote accepted

We try to discover something that might work.

Suppose we are given a positive rational $r$ such that $r^2\gt 2$. We want to produce a smaller positive rational $s$ such that $s^2\gt 2$.

We will produce $s$ by taking a little off $r$, say by using $s=r-\epsilon$, where $\epsilon$ is a small positive rational.

So we need to make sure that $(r-\epsilon)^2$ is still $\gt 2$.

Calculate. We have $$(r-\epsilon)^2-2=(r^2-2)-2r\epsilon+\epsilon^2\gt (r^2-2)-2r\epsilon.$$ If we can make sure that $(r^2-2)-2r\epsilon\ge 0$ we will have met our goal. That can be done by choosing $\epsilon=\frac{r^2-2}{2r}$. That leads to the choice $$s=r-\frac{r^2-2}{2r}=\frac{r^2+2}{2r}$$

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perfect solution, thanks Andre! –  tamefoxes Sep 17 '13 at 0:40
    
You are welcome. I hope I kind of managed to do what motivated me to answer. Finding an $s$ that works comes from a more or less natural exploration. –  André Nicolas Sep 17 '13 at 0:43
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