Negative_of_Supremum_is_Infimum_of_Negatives

Let $S$ be a subset of the real numbers $\R$. Let $S$ be bounded above. Then $(1): \quad \set {x \in \R: -x \in S}$ is bounded below $(2): \quad \displaystyle -\sup_{x \mathop \in S} x = \map {\inf_{x \mathop \in S} } {-x}$ where $\sup$ and $\inf$ denote the supremum and infimum respectively.