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In class we discussed the set $B=\{b\in\mathbb{R}|b^2>2\}$ has a supremum and infimum with respective values $\infty$ and -$\infty$.

We were given the problem of showing $sup(A)=inf(B)$ given $A=\{a\in\mathbb{Q}|a^2<2\}$ and $B=\{b\in\mathbb{Q}|b^2>2\}$. I know $sup(A)=\sqrt{2}$, but why does $inf(B)=\sqrt{2}$ with the set of rational numbers compared to the real numbers?

  • $B$ should be ${b\in\Bbb Q_{\ge 0}:b^2>2}$. – Brian M. Scott Sep 27 '20 at 00:29
  • I don't know what that distinction means. –  Sep 27 '20 at 00:41
  • $B$ should be ${b\in\Bbb Q:b^2>0\text{ and }b\ge 0}$; does that help? – Brian M. Scott Sep 27 '20 at 00:43
  • Thanks for the clarification, but barely. I appreciate your help, though. –  Sep 27 '20 at 00:47
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    The point is that if you limit yourself to non-negative rationals (or reals) in the definition of $B$, then $\inf B$ actually is $\sqrt2$: it was only the negative numbers in the original version of $B$ that loused things up and gave you an infimum of $-\infty$. – Brian M. Scott Sep 27 '20 at 00:49
  • Oh. Thanks. Are you able to recommend any supplemental texts in real analysis? We're using Abbott's Understanding Analysis but evidently it's not helping. –  Sep 27 '20 at 00:54
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    It’s probably been a good 30 years since I last taught real analysis, so I no longer have any real feel for the available texts. In particular, I’m not familiar with Abbott, so it would be difficult to suggest something really complementary. (I do know that a number of folks have recommended it for self-study.) I strongly suspect that others here could give you better advice. – Brian M. Scott Sep 27 '20 at 01:15
  • Thanks, I appreciate it. –  Sep 27 '20 at 01:16
  • You’re welcome; good luck! – Brian M. Scott Sep 27 '20 at 01:16
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    Thanks, I need it. This is what happens when a statistician is thrown into an analyst's paradise. –  Sep 27 '20 at 01:18
  • As Brian says, you should throw away the negative reals and rationals for the purpose of this question. It might help to understand it, if you draw the graph of y=x^2, for x>=0, and the horizontal line y=2, together. Now try to shade on the horizontal axis those positive real numbers whose square is less than 2, and in a different colour, those whose square is greater than 2. – Simon Nov 15 '20 at 18:56

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