Find supremum and infimum of set 3

Question

Find the supremum and infimum and study if those are maximum or minimum of this set $\left \{ x \in \mathbb{R}: x^{2} \leq 3 \right \}$

Now, this i was thought about this set:

$\sqrt3$ is an upper bound of $\left \{ x \in \mathbb{R}: x^{2} \leq 3 \right \}$ . Thus $\text{sup(S)} \leq \sqrt3$, but $\sqrt3 \in \left \{ x \in \mathbb{R}: x^{2} \leq 3 \right \}$, so $Sup=\sqrt3$ and maximum as well. Am i wrong?

Answer 1

$$x^2\le 3 \iff (x-\sqrt {3})(x+\sqrt {3})\le 0$$

$$\iff -\sqrt {3}\le x \le \sqrt {3} $$

$$\iff x\in [-\sqrt {3},\sqrt {3}] .$$

thus, the minimum is $-\sqrt {3} $ and the maximum is $\sqrt {3} $.

Remark

This is not true in $\Bbb Q $.