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 $|x_k| \le M$ 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 $x^{(K)}_j := x_{j + K}$ 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 $\varepsilon > 0$ there is $K \in \mathbb{N}$ with $|x_k - x| < \varepsilon$ for all $k \ge K$, and $(x_k)$ is Cauchy when for every rational $\varepsilon > 0$ there is $K \in \mathbb{N}$ with $|x_k - x_l| < \varepsilon$ 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 $q \mapsto \hat q$ is an embedding of ordered fields (The rationals embed densely in the reals), and as is standard we write $q$ for $\hat q$, 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 $\varepsilon$ occurring on this page is its image under this embedding.

• Rational and real $\varepsilon$ agree. Limits and Cauchy sequences of reals tests convergence against every rational $\varepsilon > 0$, and its own remark records that this loses nothing. Spelled out: suppose that for every rational $\varepsilon > 0$ there is $K$ with $|x_k - x| < \varepsilon$ 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 $\varepsilon$ with $0 < \varepsilon < \eta$, and the index $K$ belonging to that $\varepsilon$ satisfies $|x_k - x| < \varepsilon < \eta$ 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 $\varepsilon > 0$, eventually $\cdots < \varepsilon$". Proofs on this page therefore run the test with a real $\varepsilon$ 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 $|x_k - c| = |0| = 0 < \varepsilon$ for every $\varepsilon > 0$ and every $k$; it is bounded by $|c|$.

• Eventually and frequently are dual. $P$ holds frequently exactly when $\lnot P$ does not hold eventually, and $P$ holds eventually exactly when $\lnot P$ 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.