Positive Multiple of Metric is Metric

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Let $k \in \R_{>0}$ be a (strictly) positive real number.

Let $d_k: A \times A \to \R$ be the function defined as:
 * $\forall \left({x, y}\right) \in A: d_k \left({x, y}\right) = k \cdot d \left({x, y}\right)$

Then $M_k = \left({A, d_k}\right)$ is a metric space.

Proof of $M1$
So axiom $M1$ holds for $d_k$.

Proof of $M2$
So axiom $M2$ holds for $d_k$.

Proof of $M3$
So axiom $M3$ holds for $d_k$.

Proof of $M4$
So axiom $M4$ holds for $d_k$.