Choose a nonempty $X\subseteq A$ for which $|X+A|/|X|$ is minimal, and write this minimum as $K'$. Since $X=A$ is an available choice, $K'\leq K$. The <Petridis minimal-growth lemma> gives
$$
|X+A+C|\leq K'|X+C|
$$
for every finite $C$. Taking $C=(t-1)A$ and iterating yields
$$
|X+tA|\leq(K')^t|X|\leq K^t|X|
$$
for every integer $t\geq1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 129 / 1 / a / Solution

 Solution (source code)