Problem: Let S be a non-empty set of real numbers, and prove that the following statements are equivalent:
(1) If v is any upper bound of S, then u <= v (read as "u is less than or equal to v").
(2) If z < u, then z is not an upper bound of S.
(3) If z < u, then there exists s_z (read as "s sub z") in S such that z < s_z.
(4) If epsilon > 0, then there exists s_epsilon (read as "s sub epsilon") in S such that u - epsilon < s_epsilon.