Completeness, Suprema and Infima

Let $S \subseteq \mathbb{R}$ be non-empty. A number $M$ is an upper bound of $S$ if $x \le M$ for all $x \in S$. The supremum $\sup S$ is the least upper bound: the smallest number that is still $\ge$ every element. Dually, $\inf S$ is the greatest lower bound.

The defining property of the real numbers — the Completeness Axiom — is that every non-empty set of reals that is bounded above has a supremum in $\mathbb{R}$. This is exactly what the rationals lack, and it is the foundation on which limits, continuity and integration are built.

Worked example. Let $S = \left\{\, 1 - \tfrac{1}{n} : n \in \mathbb{N},\, n \ge 1 \right\} = \{\,0,\ \tfrac12,\ \tfrac23,\ \tfrac34,\ \dots\}$.

Every term is less than $1$, and the terms climb toward $1$ without ever reaching it, so $\sup S = 1$ (a least upper bound that is not attained). The smallest term is $1 - \tfrac11 = 0$, which is in $S$, so $\inf S = 0$ (attained — it is also the minimum). "Attained" and "is the sup/inf" are separate questions.