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:


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

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