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In Terence Tao's book, Additive Combinatorics, page 70, it says:

For instance, if one knows that $$ A + B \subseteq A + X $$ then one can immediately deduce that $$ A + n B \subseteq A + n X $$ for all $n ≥ 0$.

I'm really confused about why this could work. Any ideas?

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up vote 1 down vote accepted

Adding $iB+(n-i-1)X$ to the first equation:

$$A+(i+1)B + (n-i-1)X\subset A+iB+(n-i)X$$

Therefore $A+nB \subset A+(n-1)B+X \subset \cdots \subset A+nX$

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