# Thread: Prove sup S <= inf T

1. ## Prove sup S <= inf T

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.

2. Suppose the contrary, that sup S > inf T. Let $\epsilon=(\sup S-\inf T)/2$, and follow the definitions of sup and inf to derive a contradiction.