Multiple of Infimum

Theorem
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:
 * $\displaystyle \inf_{x \mathop \in T} \left({z x}\right) = z \ \inf_{x \mathop \in T} \left({x}\right)$

where $\inf$ denotes infimum.

Proof
From Negative of Infimum is Supremum of Negatives:
 * $\displaystyle -\inf_{x \mathop \in T} x = \sup_{x \mathop \in T} \left({-x}\right) \implies \inf_{x \mathop \in T} x = -\sup_{x \mathop \in T} \left({-x}\right)$

Let $S = \left\{{x \in \R: -x \in T}\right\}$.

From Negative of Infimum is Supremum of Negatives, $S$ is bounded above.

From Multiple of Supremum:
 * $\displaystyle \sup_{x \mathop \in S} \left({z x}\right) = z \ \sup_{x \mathop \in S} \left({x}\right)$

Hence:
 * $\displaystyle \inf_{x \mathop \in T} \left({z x}\right) = -\sup_{x \mathop \in T} \left({-z x}\right) = -z \ \sup_{x \mathop \in T} \left({-x}\right) = z \ \inf_{x \mathop \in T} \left({x}\right)$