Multiple of Infimum

Theorem
Let $$S \subseteq \mathbb{R}: S \ne \varnothing$$ be a non-empty subset of the set of real numbers.

Let $$S$$ be bounded below.

Let $$z \in \mathbb{R}: z > 0$$ be a positive real number.

Then $$\inf_{x \in S} \left({zx}\right) = z \inf_{x \in S} \left({x}\right)$$.