Idea for Problem 5 submitted by Simon Elimelakh

Problem 5

Denote by M the set of numbers of the type A + B 2^1/2, where A and B are integers. Prove that any neighborhood of a real number contains an infinite number of elements of the set M.

Idea

M is closed under addition and multiplication. sqrt(2) - 1 < 1/2, so its powers yield arbitrarily small elements of M. Therefore, M is dense.

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Last revised August 2003