State and prove the Knaster-Tarski Fixed-Point Theorem. Deduce the SchröderBernstein Theorem.

Show that the poset $P$ of all countable subsets of $\mathbb{R}$ (ordered by inclusion) is not complete.

Find an order-preserving function $f: P \rightarrow P$ that does not have a fixed point. [Hint: Start by well-ordering the reals.]