S = {x ∈ Q : x² < 2} Prove sup(S) = - inf(S)

Question

Define $S = {x \in Q : x^2 < 2}$. Let $a = \sup(S)$ and $b = inf(S)$. Prove that $a = -b$. (without finding $a$ or $b$)

I know $b ≤ x ≤ a \; ∀ x \in S$ and $∀ \epsilon > 0 \; ∃ y_1, y_2$ s.t. $y_1 > a - ϵ$ and $y_2 < b + ϵ$ But how do I prove $a = -b$?

Answer 1

Note your $S$ satisfies the following property: $$x\in S\Longrightarrow -x\in S.$$ Assume $\sup S>-\inf S$, then there exists a sequence $x_n\rightarrow \sup S$, and $-x_n<\inf S$.

Similar argument if $\sup S<-\inf S$.