# Section 2.8: Problem 4 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James . Munkres

Let
. Show that
iff
is finite.

If
is finite, then it can be described by a sentence
, which is true in
, and, hence, in
, implying
. If
is unbounded, then the sentence
is true in
, and, hence, in
, implying, in particular, that for any infinite
, there is
such that
, i.e.
, and
. Finally, if
is bounded but infinite, then, according to Exercise 5, there is a point
such that
is infinitely close to some
and
, implying that
, and
.