The exercise is givien as the title says, I can't see no point using that $\kappa(G) \leq \lambda(G) \leq \delta(G) \leq \Delta(G)$, but is the only inequality that I know related with the connectivity with edges or vertex.
Little addition, it is obvious that only exists two cases, $\kappa(G) = 0$ which makes the Graph void. And $\kappa(G) = 1$ so the Graph has a cut point, anyway, I can't see any further or a relation with $\lambda(G)$ besides the inequality.
Any hints?
Is it right that $\lambda$ is the minimum size of edge cut?
Then, the problem is easy. (Cosider only for $\kappa=1$).
Choose such cut vertex $v$. Let $C_1,\cdots,C_k$ be the components of $G-v$. And observe edges incident with $v$ and $C_i$.
From the fact that $k\geq 2$, we can find some $C_i$ which has at most $\Delta/2$ edges between $v$. These edges make $\lambda \leq\Delta/2$.