A Bounded Monotone Sequence of Real Numbers Converges

In the setting of The Real Line: Standing Notation and Background for Calculus, let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers, and let

be its set of terms, which is nonempty. Suprema and infima are as in The Real Line: Standing Notation and Background for Calculus §completeness.

1. (Nondecreasing case) ¶ If $a_n\le a_{n+1}$ for every $n\in\mathbb{N}$ and $A$ is bounded above, then $\sup A$ exists and $(a_n)$ converges to $\sup A$.

2. (Nonincreasing case) ¶ If $a_{n+1}\le a_n$ for every $n\in\mathbb{N}$ and $A$ is bounded below, then $\inf A$ exists and $(a_n)$ converges to $\inf A$.