Let X be a non-empty bounded subset of R and s be its supremum. If −X is the set of all −x, for x in X, prove that −s is the infimum of −X.

Question

Let X be a non-empty bounded subset of R and s be its supremum. If −X is the set of all −x, for x in X, prove that −s is the infimum of −X.

Answer 1

If s is the supremum of X, then we know ∀x in X, x ≤ s and s is the least upper bound. That is, ∀ s' such that s' is an upper bound, s ≤ s'

If -X just -x ∀x in X, then we know that if we negate both sides of the inequality above, we flip the direction of the inequality as well. In other words, if x ≤ s, we know that -x ≥ -s. We also know that -s' ≥ -s, which means that this is the greatest lower bound on this set, so -s must be the infimum.