True or false : Let $b \in \mathbb R, b>0$ and let $B:=\{x^2 :\, x \in (−b,0)\}$. We then have that $\inf(B) = 0$ and $\sup(B) = b^2$

Question

Statement:

Let $b \in \mathbb R, b>0$ and let $B:=\{x^2 :\, x \in (−b,0)\}$. We then have that $\inf(B) = 0$ and $\sup(B) = b^2$.

I'm not sure where to start with this question. Can anyone help?

Answer 1

Hint: Remember that $p = \sup B$ if and only if for every $ε > 0$ there is an $x ∈ B$ with $x > p − ε$, and $x ≤ p$ for every $x ∈ B$. And $p = \inf B$ if and only if for every $ε > 0$ there is an $x ∈ B$ with $x < p + ε$, and $x ≥ p$ for every $x ∈ B$.

It's clear that $$\forall x \in B,\, b^2 \ge x$$ But you have to show that $$ \forall \epsilon >0,\, \exists x \in B,\, b^2 - \epsilon < x $$