Sequences of reals: bounded, eventually, frequently, tails, subsequences

Throughout, $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property), The real numbers) with its order and absolute value (Order on the reals), and $\mathbb{N}$ is the set of natural numbers with its order (The natural numbers $\mathbb{N}$ (von Neumann), Order on the natural numbers).

A sequence of reals is a function $x:\mathbb{N}\to \mathbb{R}$. We write $x_k$ for the value $x(k)$, call it the $k$-th term, and write $(x_k)$, or $(x_k{)}_{k\in \mathbb{N}}$, for the function itself. The range of $(x_k)$ is the subset $\{x_k:k\in \mathbb{N}\}\subseteq \mathbb{R}$.

Let $(x_k)$ be a sequence of reals and let $P$ be a property of indices.

• $(x_k)$ is bounded if there is $M\in \mathbb{R}$ with $\mathrm{\mid }{x}_k\mathrm{\mid }\le M$ for every $k\in \mathbb{N}$; it is unbounded otherwise.
• $P$ holds eventually if there is $K\in \mathbb{N}$ such that $P(k)$ holds for every $k\ge K$.
• $P$ holds frequently if for every $K\in \mathbb{N}$ there is some $k\ge K$ for which $P(k)$ holds.
• For $K\in \mathbb{N}$, the $K$-th tail of $(x_k)$ is the sequence $x^{(K)}$ defined by $x_j^{(K)}:=x_{j+K}$ for $j\in \mathbb{N}$; it is again a sequence of reals.
• A function $n:\mathbb{N}\to \mathbb{N}$ is strictly increasing if $n_j<n_k$ whenever $j<k$. For such an $n$, the subsequence of $(x_k)$ along $n$ is the composite $x\circ n$, written $(x_{n_k}{)}_{k\in \mathbb{N}}$; it is again a sequence of reals.

Convergence and Cauchyness are not defined here. They are already fixed, for sequences of reals, by the published Limits and Cauchy sequences of reals: $(x_k)$ converges to $x\in \mathbb{R}$ when for every rational $\epsilon >0$ there is $K\in \mathbb{N}$ with $\mathrm{\mid }{x}_k-x\mathrm{\mid }<\epsilon$ for all $k\ge K$, and $(x_k)$ is Cauchy when for every rational $\epsilon >0$ there is $K\in \mathbb{N}$ with $\mathrm{\mid }{x}_k-x_l\mathrm{\mid }<\epsilon$ for all $k,l\ge K$. This page builds the toolkit for those two notions and does not restate them. A sequence converges if it converges to some real, and diverges if it does not.

• Identification of $\mathbb{Q}$ with its image. The map $q\mapsto \widehat{q}$ is an embedding of ordered fields (The rationals embed densely in the reals), and as is standard we write $q$ for $\widehat{q}$, so that a rational may be compared with a real without further comment. Limits and Cauchy sequences of reals is stated with the hat; every rational $\epsilon$ occurring on this page is its image under this embedding.

• Rational and real $\epsilon$ agree. Limits and Cauchy sequences of reals tests convergence against every rational $\epsilon >0$, and its own remark records that this loses nothing. Spelled out: suppose that for every rational $\epsilon >0$ there is $K$ with $\mathrm{\mid }{x}_k-x\mathrm{\mid }<\epsilon$ for all $k\ge K$, and let $\eta >0$ be an arbitrary real. Since $0<\eta$, density of $\mathbb{Q}$ in $\mathbb{R}$ (The rationals embed densely in the reals) supplies a rational $\epsilon$ with $0<\epsilon <\eta$, and the index $K$ belonging to that $\epsilon$ satisfies $\mathrm{\mid }{x}_k-x\mathrm{\mid }<\epsilon <\eta$ for all $k\ge K$. The converse implication is immediate, since every positive rational is a positive real. So the two formulations define the same relation, and the same two lines apply verbatim to the Cauchy condition and to any condition of the shape "for every $\epsilon >0$, eventually $\cdots <\epsilon$". Proofs on this page therefore run the test with a real $\epsilon$ wherever that is more convenient, and say so by citing this remark; nothing is smuggled in.

• Constant sequences converge. For $c\in \mathbb{R}$ the constant sequence $x_k:=c$ converges to $c$, because $\mathrm{\mid }{x}_k-c\mathrm{\mid }=\mathrm{\mid }0\mathrm{\mid }=0<\epsilon$ for every $\epsilon >0$ and every $k$; it is bounded by $\mathrm{\mid }c\mathrm{\mid }$.

• Eventually and frequently are dual. $P$ holds frequently exactly when $\mathrm{\neg }P$ does not hold eventually, and $P$ holds eventually exactly when $\mathrm{\neg }P$ does not hold frequently. So the two quantifier patterns are negations of one another applied to the complementary property, and there is no third pattern hiding between them.

• A sequence is not its range. $(x_k)$ is a function, not a set, and the range does not determine the function: the sequence $x_0=0$, $x_k=1$ for $k\ge 1$ and the sequence $y_0=y_1=0$, $y_k=1$ for $k\ge 2$ have the same range $\{0,1\}$, yet they differ at $k=1$, so they are different sequences. Order and repetition are part of the data and the range forgets both. Boundedness, on the other hand, depends only on the range.

• Indexing. Sequences here start at $k=0$ because $\mathbb{N}$ contains $0$ (The natural numbers $\mathbb{N}$ (von Neumann)). Texts that index from $1$ describe the same objects up to a shift, and the shift changes nothing about convergence, by Convergence depends only on the tail.