Does your professor require you to use the full definition of the supremum? (i.e. the supremum about being the lowest greater bound?)
This is from Fitzpatrick's Advanced Calculus. I am not very experienced with proofs so could someone tell me if this is a correct way of doing it? Thanks!
Claim: If is a set of real numbers that is bounded above and is a nonempty subset of , then .
Proof: Let be a set of real numbers that is bounded above and a nonempty subset of . Take an arbitrary element in . Since , is also in . Since is bounded above, by the completeness axiom we know there is a number such that , and there is no number less than that is an upper bound for . Since is in , is also an upper bound for . So we must have , or .
Chiro is correct to refer you to your professor - there are probably certain things that you need to make sure you include.
I would write a lot less to prove this claim: Let . Then since B is a subset of S, , so . Since x was an arbitrary element of B, .
But again, your professor might want more.