Determine Supremum and Infimum of a Set

Determine the supremum and infimum of the set $\left\{ \frac{1}{n} : n \in \mathbb{N} \right\}$.

In this problem, you are tasked with finding the supremum and infimum of the set of numbers represented by the reciprocals of natural numbers. Conceptually, this involves understanding the definition and significance of supremum and infimum within the context of real numbers. The supremum of a set is the least upper bound, the smallest number that is greater than or equal to every element in the set, while the infimum is the greatest lower bound, the largest number that is less than or equal to every element in the set.

For the given set, you'll notice that as n becomes very large, the terms in the set get smaller and closer to zero, which will play a key role in determining these bounds. When tackling such problems, it is important to remember that the supremum does not have to be an element of the set, but must be greater than or equal to all elements of the set. Similarly, the infimum may or may not be part of the set, but must be less than or equal to every element.

This problem not only reinforces the understanding of these concepts but also encourages a methodical approach to analyzing sets and their boundaries. Pay close attention to the behavior of the set elements as n varies and apply the definitions of supremum and infimum to deduce the answer.