Suppose the contrary, that sup S > inf T. Let , and follow the definitions of sup and inf to derive a contradiction.
Let S and T be non-empty subsets of R, and suppose that for all s ε S and t ε T, we have s <= t. Prove that supS <= infT.
Here's what I have for it:
Since s ε S and s ε T, supT is an upper bound for S.
Since supS is the least upper bound, sup S <= sup T.
How does that look? Help is greatly appreciated. Thanks.