K-Connectivity Implies Lesser Connectivity

Theorem
Let $G = \struct {V, E}$ be a $k$-connected graph.

Then $G$ is $l$-connected for all $l \in \Z : 0 < l < k$.

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

Then:


 * $\card V > k > l$
 * $G$ is connected
 * If $W$ is a vertex cut of $G$, then $\card W \ge k > l$ so $\card W \ge l$.