Theorem: For every non empty set E of real numbers that is bounded above there exists a unique real number sup(E) such that 1. sup(E) is an upper bound for E. 2. if y is an upper bound for E then y [itex]\geq[/itex] sup(E). Prove: [itex]sup(A\cap B)\leq sup(A)[/itex] I can show a special case of this, if [itex]A\cap B=\emptyset [/itex], then [itex]sup(A\cap B)\leq sup(A)[/itex]. Nothing is less than something, right? Now here's my problem... Beyond the trivial case, all I have been able to do is draw pictures of sets on a number line. The pictures make the inequality really obvious, but I don't think that pictorial intuition counts as a real proof. Could anyone give me a pointer on how to set up a real proof? thanks!