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 $
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.
