Let $\kappa(G)$ be the connectivity of a graph $G$, $|V| = n$ and $|E|=m$.

For any graph $G$, prove that if $m \geq n-1$ then

$$\kappa(G) \leq \lfloor \frac{2m}{n} \rfloor$$ What I know is that $\kappa(G) \leq \lambda(G) \leq \delta(G)$ where $\lambda(G)$ is the edge connectivity and $\delta(G)$ is the minimum degree.

So maybe I could show that this is an upper bound on $\delta(G)$?

Also I know that from the handshaking lemma that the sum of degrees of the vertices is twice the number of edges; that is,

$\sum\limits_{v \in V}d(v) = 2|E|$

Any hints or ideas would be great.

Try to think of an interpretation of $\frac{2m}{n}$ and then relate it to the minimum degree $\delta(G)$.

Hint: The interpretation has to do with degree.

Note that $\frac{2m}{n}=\frac{\sum_{v\in V}\deg(v)}{n}$ and you can think about the right hand side as the average degree of vertices in $G$.