Let S be a non-empty set of real numbers which is bounded above, and suppose that x = sup(S). - (a) Prove that for all € > 0 there exists s € S so that s ≤ (x − €, x].

1. (a) (b) (c) (d) Prove or disprove that, for any universal set U and predicates P and Q, [3x = U, P(x) ^ Q(x)] → [3r EU, P(x)) ^ (3x = U, Q(x))] Prove or disprove that, for any universal set U and predicates P and Q, [3r EU, P(x)) ^ (3xU, Q(x))] → [r U, P(x) ^ Q(x)] Prove or disprove that, for any universal set U and predicate P [3r € U, P(x)] → [Vr € U, P(x)] Prove or disprove that, for any universal set U and predicate P [VxU, P(x)] → [3r € U, P(x)]