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