Halin's theorem and Mader's theorem

Prove the following :

• Halin's Theorem : (Easy) There is a node of degree $k$ in any edge-minimal $k$-vertex-connected graph.

• Mader's Theorem : (Hard) In any edge-minimal $k$-vertex-connected graph every cycle has a node of degree $k$.

Note that Mader's Theorem implies Halin's Theorem.