# 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
.