# 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
.