Totally Ordered Set is Lattice

Theorem
Every totally ordered set is a lattice.

Proof
Let $$\left({S, \preceq}\right)$$ be a totally ordered set.

Then we have:


 * $$\forall x, y \in S: x \preceq y \or y \preceq x$$
 * $$\forall x, y \in S: x \preceq y \implies \sup \left\{{x, y}\right\} = y \and \inf \left\{{x, y}\right\} = x$$
 * $$\forall x, y \in S: y \preceq x \implies \sup \left\{{x, y}\right\} = x \and \inf \left\{{x, y}\right\} = y$$

Thus the conditions for $$\left({S, \preceq}\right)$$ to be a lattice are fulfilled.