How to Prove the Existence of a Cycle Subgraph for k-Connected Graphs?

[andreass]

How to prove that for every k-connected graph (k>=2) with at least 2*k vertices, there exists subgraph, which is cycle with at least 2*k vertices?
Ok, it’s obvious for k=2. It looks something like cycle with or without some other edges:

But I've no ideas how to prove it for k>2
Any hints?

 

[Outlined]

Is this your definition of k-connected?:
http://mathworld.wolfram.com/k-ConnectedGraph.html

 

[andreass]

Yes, it's the same.
But in my opinion "better" definition is:
Graph is k-connected if and only if it contains k internally disjoint paths between any two vertices