K-Connectivity Implies Lesser Connectivity

Theorem
If a graph $G$ is $k$-connected, then $G$ is $\ell$-connected for all $\ell \in \Z : 0 < \ell < k$.

Proof
Suppose that $G$ is $k$-connected. Then


 * $\left|{V \left({G}\right) }\right| > k > \ell$
 * $G$ is connected
 * If $W$ is a vertex cut of $G$, then $\left|{W}\right| \geq k > \ell$ so $\left|{W}\right| \geq \ell$.