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.