This is exact Theorem 6 in

M. B. Nathanson, Sumsets contained in infinite sets of integers, J. Combina. Theory, Series A, 28 (1980), 150 – 155.

Also I tend to use the symbol for the additive group and for the multiplicative group of complex numbers with absolute value 1 (of course these groups are isometrically isomorphic but it helps to distinguish additive and multiplicative notation) ]]>