Negative_of_Infimum_is_Supremum_of_Negatives

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