I started trying to prove by induction on $|E|$, using the characterization of trees, but cannot get any further. Is this a good way to proceed or shall I try something else?
The start of my induction proof had that $G$ is a graph $|E|>|V|- 1$. Hence, $G$ is not acyclic, thus it must contain a cycle $C$.
Unsure how to move on from here
For each component of a forest, we have that $E_i=V_i-1$, $E=\sum E_i, V=\sum V_i$. We also know, that for a tree, any addition of an edge adds at least one cycle. Thus, take each connected component of your graph and remove $c_i$ edges, thus that your graph is now a forest with each tree satisfying: $$E_i-c_i=V_i-1$$ Now add all the $c_i$ edges back, thus creating at least: $$c=\sum c_i = \sum E_i-V_i+1=E-V+1$$ cycles. $\blacksquare$