Least greatest proofs

For a set of numbers \(X\), how do you formalize “there is a greatest \(X\)” or “there is a least \(X\)”?

Prove or disprove: There is a least prime number.

Prove or disprove: There is a greatest integer.

Approach 1, De Morgan’s and universal generalization:

Approach 2, proof by contradiction:

Extra examples: Prove or disprove that \(\mathbb{N}\), \(\mathbb{Q}\) each have a least and a greatest element.

Least greatest proofs

For a set of numbers \(X\), how do you formalize “there is a greatest \(X\)” or “there is a least \(X\)”?

Prove or disprove: There is a least prime number.

Prove or disprove: There is a greatest integer.

Approach 1, De Morgan’s and universal generalization:

Approach 2, proof by contradiction:

Extra examples: Prove or disprove that \(\mathbb{N}\), \(\mathbb{Q}\) each have a least and a greatest element.