Ruzsa covering lemma

Statement

The Ruzsa covering lemma states the following:

Let $A$ and $S$ be finite subsets of an abelian group with $S$ nonempty, and let $K$ be a positive real number. Then if $|A+S| \le K|S|$, there is a subset $T$ of $A$ with at most $K$ elements such that $A \subseteq T+S-S$.

This lemma provides a bound on how many copies of $S-S$ one needs to cover $A$, hence the name. The proof is essentially a greedy algorithm: