Totally Ordered Set is Lattice

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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.

$\blacksquare$


Sources