Multiple_of_Infimum

Let $T \subseteq \R: T \ne \varnothing$ be a non-empty subset of the set of real numbers $\R$. Let $T$ be bounded below. Let $z \in \R: z > 0$ be a (strictly) positive real number. Then $\displaystyle \map {\inf_{x \mathop \in T} } {z x} = z \, \map {\inf_{x \mathop \in T} } x$ where $\inf$ denotes infimum.