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Section 2.8: Problem 2 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
(a) Let and . Then is also a binary relation on ; show that .
(b) Let . Recall that is said to converge to iff for every there is some such that for all , . Show that this is equivalent to the condition: for every infinite .
(c) Assume that and converges to for . Show that converges to and converges to .
(a) Consider the following sentence: where the quantifier symbol is defined in Exercise 21 of Section 2.2, and the ternary connective symbol is defined in Exercise 13-A of Section 1.5. The sentence is true in and , therefore, defines a function on .
(b) According to (a), defines a function on . Let be the relation defining in ( iff ). If converges to , then for every , , and some , the sentence is true in , and, hence, in . In particular, for an infinite , this means for every , , i.e. . Now, if for every infinite , then for every , , the sentence is true in (since we can take any infinite ), and, hence, in , implying that converges to .
(c) According to (b), for every infinite , , and , implying converges to .