Final answer:
The statement is false; closed sets are not necessarily bounded. Closed sets contain all their limit points, while bounded sets are contained within a finite interval. Examples include integers as a closed but unbounded set and an open interval as bounded but not closed.
Step-by-step explanation:
The statement that a set that is closed is also necessarily bounded is false. While closedness and boundedness are related concepts in mathematics, they do not imply each other. A set is considered closed if it contains all its limit points, meaning that any sequence of points within the set that converges will converge to a point within the set itself. On the other hand, a set is bounded if there is a real number such that all the elements of the set have a lesser absolute value than this number.
An example of a closed yet unbounded set is the set of all integers, which contains all its limit points but extends infinitely in both the positive and negative directions, and hence it is not bounded. Meanwhile, an example of a bounded yet not closed set could be the open interval (0, 1), which does not include its limit points of 0 and 1. Therefore, while there can be sets that are both closed and bounded, being closed does not automatically mean a set is also bounded.