TITLE: Converse to a proposition on finite sets of real numbers
QUESTION [6 upvotes]: Every finite set of reals has the property that every non-empty subset of it has a maximum and a minimum. Does this property characterize finite sets of reals? That is, is it the case that if $S$ is a subset of the real numbers where every non-empty subset of $S$ has a maximum and a minimum, then $S$ is finite?

REPLY [5 votes]: True. Suppose $S$ is an infinte set. If it is unbounded it is clear that there is a subset with no maximum or a subset no minimum. Suppose it is bounded. Let $(x_n)$ be  sequence of distinct  points of $S$. Then either there is a subsequence which increases to some limit $x$ or there is a subsequence which decreases to some limit $x$. In the first case $(x_{n_k})$ has no maximum and in the second case $(x_{n_k})$ has no minimum.