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Section 2.8: Problem 3 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 be one-to-one, where . Show that if but , then .
According to Exercise 2(a), . Suppose . cannot be in , because the sentence is true in , and, hence, in . But then, is in , and there is such that the sentence is true in , and, hence, in , implying that .
Note. This exercise also implies that for any , . Indeed, consider such that , for which is true in , and, hence, in . And if , then .