Minimal size of a sumset over $\mathbb{F}_p$

Question

Let $A, B \subseteq \mathbb{F}_p$ ($p$ a prime). How to show that $|A+B| \ge \min\{p, |A|+|B|-1\}$?

Since $\mathbb{F}_p$ has only $p$ elements, $\forall S \subseteq \mathbb{F}_p, |S| \ge \min\{p, |S|\}$. Then it's enough to show that $|S| \ge |A|+|B|-1$, but I'm not sure how.

Answer 1

What you are asking about is the Cauchy-Davenport theorem. It is certainly non-trivial, so you may want to look up the proof.