Given an graph $G$ and two, randomly chosen, non-adjacent nodes, let's call them $s$ and $t$, we make the following two notations:
$P_L(s,t;G)$ = the maximum number of internally disjoint paths (paths that don't have any node in common, except s and t) with a length at most $L$ (L ranges from 1 to |G|).
$K_L(s,t;G)$ = the minimum size of a set of nodes (s and t not included in the sets) with this property - if we remove the set of nodes, the result is that there are no more paths of length at most $L$, from $s$ to $t$.
I have to prove the inequality $$ P_L(s,t;G) \le K_L(s,t;G) $$ and specify the values of $L$ (L ranging from 2 to |G|) for which the relation satisfies the equality too, for any chosen graph G and nodes s,t.
The first idea that comes to my mind is to start a discussion based on the values of $L$ and try to prove the relation by induction. Any ideas how?