Cauchy-Davenport Theorem

Let  be a nonnegative integer and let , ...,  be nonzero elements of  which are not necessarily distinct. Then the number of elements of  that can be written as the sum of some subset (possibly empty) of the  is at least . In particular, if , then every element of  can be so written.

REFERENCES:

Martin, G. "Dense Egyptian Fractions." Trans. Amer. Math. Soc. 351, 3641-3657, 1999.

Vaughan, R. C. Lemma 2.14 in The Hardy-Littlewood Method, 2nd ed. Cambridge, England: Cambridge University Press, 1997.