Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers: $$|A+qA|\ge (q+1)|A|-C_q.$$ This is a problem from Miklos Schweitzer 2013. I tried to use the fact $|A+B|\ge |A|+|B|-1$ but no avail. Here $A+qA=\{a+qa^{\prime} | a,a^{\prime}\in A \}$. Any help? I am looking for an elementary solution.
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