Defiinition of Sup/Inf from Quiz 1 Study Guide

I hope this question isn't too dumb… but for the definition of supremum and infimum do we need to know simply that a supremum is a least upper bound and an infimum is the greatest lower bound? Or do we also need to have memorized their criteria?

Not dumb. I think we will definitely need to know the explicit definitions using notation.

I agree. I am working on remembering it directly from the book:

A real number $s$ is the least upper bound of a set $A \subseteq \mathbb{R}$ if:

1. $s$ is an upper bound for $A$
2. if $b$ is any upper bound for $A$, then $s \leq b$

Then we can make a definition for infimum as follows...

A real number $t$ is the greatest lower bound for a set $A \subseteq \mathbb{R}$ if:

1. $t$ is a lower bound for $A$
2. if $c$ is any lower bound for $A$, then $t \geq c$

I agree, knowing the definition should be nearly the same as knowing the symbols and inequalities. Knowing the meaning of the the definition in mathematical terms will also make your proofs more seamless.

I think it's also worth noting that the definition of supremum we used in class was a little different, though I would still know both.

From the notes on 8/21:

If $\beta = \sup{S}$,

1. $\beta$ is an upper bound: $\beta \geq x ,\; \forall x \in S$.
2. $\beta$ is the smallest such; i.e. If $b < \beta$, then $\exists \, y \in S$ such that $y > b$.

I can't remember, but I think the book may have stated #2 as a consequence of the supremum, but not actually as part of the definition.

I think knowing it directly from the book is the way to go! Like Dr. McClure said..."now is not the time to express yourself..."

Ok, now I am questioning myself because it seems as though a lot of my definitions from class differ a bit from the book definition..

Go with the book definition so the teacher can't complain! The definitions I give in class might look a bit different but should be equivalent.

Looks good, except for the first line of the last part. You wrote, "A real number $t$ is the greatest upper bound for a set $A⊆R$ if:
$t$ is a lower bound for $A$
if $c$ is any lower bound for $A$, then $t≥c$" I'm pretty sure you meant lower bound

You are so right! Thanks, I fixed it...