IMC 2012 Problem 10

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 $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$ . Prove that $$|\underbrace{A+A+\dots+A}_k| \le c^k |A|$$ for every positive integer $k$ .

Proposed by Przemyslaw Mazur, Jagiellonian University.