Product of two subsets of a group

Consider $G$ a finite group, with $|G|=n$. I'm asking myself if, and why, is true that:

if $A$ and $B$ are two subsets (not necessarily subgroups) such that $|A|+|B|> n$ then $AB=G$.

I can't show this, but can't find a counterexample neither.

Thm. If $G$ is a finite group, $A, B\subseteq G$, and $|A|+|B| > |G|$, then $AB=G$.

Proof. (Contrapositive)

We assume that $AB\neq G$ and prove that $|A|+|B|\leq |G|$.

Choose a group element $g\notin AB$. Saying that $g\notin AB$ is saying that $g=ab$ with $a\in A$ and $b\in B$ is impossible. Hence $a^{-1}g=b$ for $a\in A$ and $b\in B$ is impossible. This shows that the set $A^{-1}g:=\{a^{-1}g\;|\;a\in A\}$ is a subset of $G$ that is disjoint from $B$. Hence $|G|\geq |A^{-1}g|+|B|=|A|+|B|$. $\Box$