Existence of Upper Bound of Finite Subset of Directed Set

Theorem
Let $\struct {\Lambda, \preceq}$ be a directed set.

Let $\FF \subseteq \Lambda$ be a finite subset.

Then $\FF$ has an upper bound in $\struct {\Lambda, \preceq}$.

Proof
We induct on the cardinality of $\FF$.

Let $n \in \Z_{\ge 0}$.

For all $m \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * if $\FF \subseteq \Lambda$ has $\card \FF = n$, then $\FF$ has an upper bound.

Basis for the Induction
The case $n = 0$ is vacuously true.

If $n = 1$ and $\FF = \set \lambda$, $\lambda$ is clearly an upper bound for $\FF$ in $\struct {\Lambda, \preceq}$.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * if $\FF \subseteq \Lambda$ has $\card \FF = k$, then $\FF$ has an upper bound.

Then we need to show:
 * if $\FF \subseteq \Lambda$ has $\card \FF = k + 1$, then $\FF$ has an upper bound.

Induction Step
This is our induction step.

Let $\FF = \set {\lambda_1, \ldots, \lambda_{k + 1} }$.

Then $\GG = \set {\lambda_1, \ldots, \lambda_k}$ has $\card \GG = k$, and so there exists $\lambda \in \Lambda$ such that:


 * $\lambda_i \preceq \lambda$ for each $1 \le i \le n$.

From the definition of a directed set, there exists $\lambda_\ast \in \Lambda$ such that:


 * $\lambda \preceq \lambda_\ast$

and:


 * $\lambda_{k + 1} \preceq \lambda_\ast$

Then we have, from transitivity:


 * $\lambda_i \preceq \lambda_\ast$ for each $1 \le i \le k + 1$.