I'm having real trouble with this practice question: 

Show that 

$$\tau(G) ≤ \frac{|E(G)| + 1}{2}$$ 
for every connected graph G.

What properties of a connected graph entail this inequality? Does it tell us something about the number of vertices?

So if a graph is connected, every vertex has at least an edge incident to it. So $|V(G)|$ is at most $|E(G)|+1$, because the connected graph with smallest $|E(G)|$ is a tree with two leaves and all other vertices of degree 2. Is it that you only need half the vertices because an edge connects two vertices, and so since we have an upper bound for $|V(G)|$ we have an upper bound for $\tau(G)$? Am I going in the right direction?

Can anyone help? Thanks!