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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 .