Completeness of the Real Numbers for Sequences: Monotone Convergence, the Bolzano-Weierstrass Theorem and Cauchy Sequences

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let convergence be as in Convergent Sequences of Real Numbers §converges; monotone, bounded and Cauchy sequences as in Monotone Sequences §monotone, Bounded Sequences of Real Numbers §bounded and Cauchy Sequences of Real Numbers §cauchy; and subsequences as in Subsequences §subsequence.

¶ $1/n\to0$.

¶ If $r\in\mathbb{R}$ and $|r|<1$, then $r^{n}\to0$.

Let $(a_{n})$ be a sequence in $\mathbb{R}$.

¶ If $(a_{n})$ is nondecreasing and bounded above, then $a_{n}\to\sup\{a_{n}:n\in\mathbb{N}\}$; if it is nonincreasing and bounded below, then $a_{n}\to\inf\{a_{n}:n\in\mathbb{N}\}$.

¶ If $(a_{n})$ is bounded, then it has a convergent subsequence.

¶ $(a_{n})$ is convergent if and only if it is a Cauchy sequence.