Is my proof for set boundedness correct?
Proposition:
Let $A\subseteq \mathbb{R}.$ Then $A$ is bounded if and only if $\exists{K}\gt{0}$ s.t $\forall{a}\in{A}, |a|\leq{K}$.
Proof:
Suppose that $A$ is bounded. Then there exists $M\in \mathbb{R}$ and $m\in\mathbb{R}$ such that $m\leq{a}\leq{M}$. Take $K=\text{max}(|M|,|m|,1)$ so that $K\gt{0}$. Therefore $-K\le{a}\le{K}$ and thus, $|a|\le{K}$.
Conversely, assume that $\exists{K}\gt{0}$ s.t $\forall{a}\in{A}, |a|\leq{K}$ holds. Then we have that $-K\leq{a}\leq{K}$. Therefore $-K\leq{a}$ for all $a\in{A}$ so $A$ is bounded below and $a\leq{K}$ for all $a\in{A}$ so $A$ is bounded above. Thus, $A$ is bounded.