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Yet again struggling.

Find $\sup A$ and $\inf A$ where $A$ is the set defined by:

(a) $A=\{x∈\mathbb{Q}:x^{2} −x<1\}$

(b) $A=\{x∈\mathbb{R}:x^3 −x\le6\}$

My answers:

(a) $\sup A= ?\quad\inf A=\frac12$

(b) $\sup A=2 \quad\inf A=-\infty$

Please correct me or put me on the right track — especially for $\sup A $

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  • $\begingroup$ There isn't a rational number closest to $\frac{1+\sqrt{5}}{2}$ $\endgroup$ Commented Nov 29, 2014 at 21:02
  • $\begingroup$ thank you- edited it-i do understand why $\endgroup$ Commented Nov 29, 2014 at 21:07

1 Answer 1

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I will assume that (in both cases) you are supposed to find the real supremum or infimum, if it exists.

You are spot on for part (b), as your set $A$ there is simply the set of all real numbers no greater than $2.$

For part (a), I'm not sure what went wrong. The set $A$ turns out to be the set of all rational numbers strictly between $\frac{1-\sqrt5}2$ and $\frac{1+\sqrt5}2.$ Since the rationals are dense in the reals, then $\inf A=\frac{1-\sqrt5}2$ and $\sup A=\frac{1+\sqrt5}2.$ If you're looking for a rational supremum/infimum, though, you're on a fruitless quest.

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