IMC 2012 · Problem 5

Let $c \ge 1$ be a real number. Let $G$ be an abelian group and let $A \subset G$ be a finite set satisfying $|A + A| \le c|A|$, where $X + Y := \{x + y \mid x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that

$$|\underbrace{A + A + \dots + A}_{k \text{ times}}| \le c^k |A|$$

for every positive integer $k$. (Plünnecke's inequality)