TITLE: If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
QUESTION [6 upvotes]: Let $G$ be a finite group. Suppose that $A$ and $B$ are to subsets of $G$. If $|G|<|A|+|B|$ prove that $$G=AB.$$

REPLY [9 votes]: Let $g \in G$. Note that $|gB^{-1}| = |B|$ so $|A| + |gB^{-1}| > |G|$; in particular, $A\cap gB^{-1} \neq \emptyset$. Let $a \in A\cap gB^{-1}$, then $a = gb^{-1}$ for some $b \in B$. Therefore $g = ab \in AB$ so $G = AB$.