Section 2.8: Problem 5 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
(Bolzano–Weierstrass theorem) Let be bounded and infinite. Show that there is a point that is infinitely close to, but different from, some member of . Suggestion: Let with one-to-one; look at for infinite .
As suggested, consider a one-to-one function (there exists one as is infinite). Then, according to Exercise 2(a), . In fact, , as the sentence is true in , and, hence, in . Moreover, is bounded, as for some , the sentence is true in , and, hence, in . Suppose is infinite, and let . Then, , i.e. , and, according to Exercise 3, , i.e. . Overall, we have , , and .