SupremumofSetNoverNplus1

Prove that the supremum of the set {$\displaystyle \frac{n}{n+1}$ | $n \in \mathbb{N}$} is 1.

In this problem, you are asked to find the supremum, or least upper bound, of a set of real numbers expressed in the form $\displaystyle \frac{n}{n+1}$, where $n$ is a natural number. Solving this problem involves understanding the behavior of the sequence as $n$ tends toward infinity. The general strategy is to evaluate the limit of $\displaystyle \frac{n}{n+1}$ as $n$ increases, reaching an understanding of how the terms in the set behave asymptotically toward a particular value.

The notion of supremum is central to real analysis as it encapsulates the idea of the boundary or edge of a set's values, where no larger value exists within certain restrictions. However, it emphasizes that a supremum might not be an actual member of the set but simply the smallest bound that no element in the set exceeds. For this set, each term is less than 1 yet approaches 1 as $n$ becomes large, making 1 a candidate for the supremum.

Logically, this also introduces concepts associated with bounded sets and limits, which help demonstrate completeness in the real number system. By firmly understanding these concepts, one can efficiently show that 1 is indeed the least upper bound, illustrating the principle that while sequence terms might never reach a particular value, they can get arbitrarily close to it, justifying 1 as the supremum of the given set.