Suppose S is a subset of R. Then diam(S) = supS − inf S.
Proof: First, if S = ∅, then inf S = ∞, supS = −∞ and diam(S) = sup∅ = −∞,
so the proposition is true by definition (B.7.1). Second, if sup S = ∞, pick
b ∈ S. Then, for each p ∈ R+, there exists some a ∈ S with a > b + p, so that...