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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?

1 Answers 1

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