Bounded Sequences of Real Numbers

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(a_{n})$ be a sequence in $\mathbb{R}$.

¶ $(a_{n})$ is bounded above, respectively bounded below, if the set $\{a_{n}:n\in\mathbb{N}\}$ of its terms is bounded above, respectively bounded below, as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §bounded; and $(a_{n})$ is bounded if there is $M\in\mathbb{R}$ with $|a_{n}|\le M$ for every $n\in\mathbb{N}$.