Find $\sup$ and $\inf$ of $A$ and justify

Question

$$A=\left \{x\in \Bbb R :x<\dfrac{2}{x} \right \}.$$

a) $\sup A = -1$ since $\max A=-1;$

b) $\inf A$ does not exist since $A$ is not bounded below.

Is this the only justifications? Can anyone check if this is correct?

Answer 1

Hint:

If $A:=\{x\in\mathbb R\mid x<\frac2{x}\}$ then $$A=(-\infty,-\sqrt2)\cup(0,\sqrt2)$$