Conventions for sequences: indexing, eventually, $\lim$, and rational $\varepsilon$

This page fixes a handful of conventions. None of them is deep, and each is the kind of thing that silently causes trouble when it is left implicit.

Indexing from $0$ or from $1$ makes no difference to convergence. Sequences here are functions on $\mathbb{N}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences), and $\mathbb{N}$ contains $0$, so the first term is $x_0$. A text that writes $(x_k)_{k \ge 1}$ is describing the first tail of such a function. By Convergence depends only on the tail a sequence and any of its tails converge to exactly the same limits and are Cauchy together, so every statement on this page about convergence, about limits and about the Cauchy condition reads the same under a shift of the indexing. That is a claim about those statements and not about every sentence here: the starting index is genuinely visible wherever an index is named, as in the base case $n_0 \ge 0$ of A strictly increasing index map satisfies $n_k \ge k$, in the $0$-th tail used in the proof of Convergence depends only on the tail, and in the recursion $s_0 = 1$ that builds the witness of FALSE: every bounded sequence converges. In a formula it shows up as the shift in $1/(k+1)$, written that way here purely so that no term is undefined at $k = 0$.

"Eventually" is the only weakened quantifier pattern used for hypotheses. A property holds eventually when it holds for all indices from some point on, and frequently when, for every index $K$, it holds at some index $k \ge K$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences). The two are negations of each other applied to the complementary property: $P$ holds frequently exactly when $\lnot P$ does not hold eventually. "Frequently" is defined in that cofinal form, and the cofinal form is what every use of the word here means, because it is the literal negation of "eventually" and needs nothing further. The familiar reading "at infinitely many indices" is equivalent to it, and the equivalence is available in this library rather than merely plausible: finiteness is defined (Finite, countably infinite, countable, uncountable), Every subset of an at most countable set is at most countable proves that a subset of $\mathbb{N}$ is finite when it is bounded above and countably infinite when it is not, and The pigeonhole principle on $\mathbb{N}$ proves that $\mathbb{N}$ is equinumerous with no natural number, so those two cases exclude one another. A set of indices is therefore cofinal exactly when it is infinite. The cofinal form is nevertheless the one taken as the definition here, because it is the literal negation of "eventually" and calls on no counting theory at all. Because of Convergence depends only on the tail, a hypothesis of the form "for all $k$" whose conclusion concerns only convergence may be weakened to "eventually" at no cost, and two of the three hypotheses of that shape on this page are stated in the weaker form: the comparison $x_k \le y_k$ of Limits preserve non-strict inequalities and the bracketing $a_k \le x_k \le b_k$ of The squeeze theorem. The third is deliberately left at every index, the strict comparison $x_k < y_k$ of FALSE: limits preserve strict inequalities: that claim is false, and stating its hypothesis at every index makes it as strong as it can be, so that the refutation defeats the strongest form rather than a weakened one. The remaining hypotheses are stated at every index because they are not of that shape. Some are needed at every index for the statement to typecheck at all, such as $y_k \ne 0$ in Algebra of limits: sums, scalar multiples, products and quotients and $x_k > 0$ in For positive terms, null and divergence to $+\infty$ are reciprocal, where a missing index would leave $1/y_k$ or $1/x_k$ undefined; and the bound $|y_k| \le M$ of A null sequence times a bounded sequence is null is not weakened because, as recorded there, an eventual bound is already a bound. "Frequently" is defined here for completeness and is used in no hypothesis on this page.

$\lim$ is written only after uniqueness is known. The notation $\lim_k x_k$ presupposes that at most one real can be the limit, which is A sequence has at most one limit. Before that lemma, the correct phrasing is "$(x_k)$ converges to $x$", a relation between a sequence and a real, not a function of the sequence. The same discipline is applied to $\sup$ and $\inf$ elsewhere in the library (Conventions: $\sup \emptyset$, unbounded sets, and the extended reals): a notation is introduced only once the object it names has been shown to exist and to be unique.

Convergence is tested against rational $\varepsilon$. The published Limits and Cauchy sequences of reals quantifies over rational $\varepsilon > 0$ rather than real $\varepsilon > 0$. That is a deliberate feature of the construction of $\mathbb{R}$, where the rationals are available before the reals are complete, and it loses nothing: below any real $\eta > 0$ lies a positive rational (The rationals embed densely in the reals), so the two formulations define the same relation. The argument is written out once, in the remarks of Sequences of reals: bounded, eventually, frequently, tails, subsequences, and proofs on this page cite it whenever a real $\varepsilon$ is more convenient than a rational one. Rationals are identified with their images in $\mathbb{R}$ under the ordered-field embedding, again as recorded in Sequences of reals: bounded, eventually, frequently, tails, subsequences.

A sequence is not its range, but boundedness only sees the range. $(x_k)$ is a function; the set $\{x_k : k \in \mathbb{N}\}$ forgets order and multiplicity. The sequence is bounded, in the sense of Sequences of reals: bounded, eventually, frequently, tails, subsequences, exactly when its range is a bounded subset of $\mathbb{R}$ in the sense of Lower bound, bounded below, bounded set. Both directions rest on the equivalence $|x| \le M \iff -M \le x \le M$: if $|x| \le M$ then $x \le |x| \le M$ and $-M \le -|x| \le x$, using $-|x| \le x \le |x|$ (Basic properties of the absolute value) together with the fact that $a \le b$ and $-b \le -a$ are the same assertion, both saying that $b - a$ is positive or zero (Ordered field); conversely, if $-M \le x \le M$ then $|x|$, being $x$ or $-x$ by the definition of the absolute value (Order on the reals, Absolute value in an ordered field), is $\le M$ either way. Given a bound $M$ with $|x_k| \le M$ for all $k$, the range is bounded below by $-M$ and above by $M$. Conversely, given $\ell \le x_k \le u$ for all $k$, put $M := \max\{|\ell|, |u|\}$, which exists because a nonempty finite list of reals has a maximum (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set); then $x_k \le u \le |u| \le M$ and $-M \le -|\ell| \le \ell \le x_k$, so $-M \le x_k \le M$ and hence $|x_k| \le M$ for every $k$, by transitivity of $\le$ (Ordered field).

Boundedness is not alone in this. Any property of $(x_k)$ that is defined purely from the set of values is settled by the range for the same trivial reason: bounded above, bounded below, and the supremum or infimum of the range when it exists (Lower bound, bounded below, bounded set, Complete ordered field (least-upper-bound property)) are read off the range by definition, so permuting or repeating terms cannot change them. Convergence is not of that kind, and that is the contrast worth drawing: the sequence $z_0 = -1$ with $z_k = 1$ for $k \ge 1$ and the alternating sequence of FALSE: every bounded sequence converges have the same range $\{-1, 1\}$, yet the first converges to $1$, being constant from index $1$ on (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Convergence depends only on the tail), while the second does not converge at all. So knowing the range settles boundedness and does not settle convergence. That is the whole of what is claimed here; each other property of a sequence has to be looked at on its own, and this page does not survey them.

"Diverges" means "does not converge". Divergence is the plain negation of convergence, so an oscillating bounded sequence diverges. Divergence to $+\infty$ or $-\infty$ (Divergence to $+\infty$ and to $-\infty$) is a strictly stronger and entirely separate condition, and the arrow in $x_k \to +\infty$ is an abbreviation, not an equation: this library never writes $\lim_k x_k = +\infty$, because $+\infty$ is not a real number.

What is deliberately absent from this page. Monotone sequences, the monotone convergence theorem, the Bolzano-Weierstrass theorem and the completeness of $\mathbb{R}$ in the Cauchy sense are none of them treated here. They are the subject of the next page of this track, Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness, which comes later in the reading order; nothing on this page may be taken to have proved any of them. The last of the four is the one exception worth stating: for the $\mathbb{R}$ this page works in, every Cauchy sequence of reals converges to a real is already proved elsewhere in the library, by The reals are complete on the Cauchy-construction page, and Every convergent sequence is Cauchy records where that leaves the two directions. The monotone convergence theorem and the Bolzano-Weierstrass theorem have no such exception: neither is available at this point in the reading order, and both are proved on that next page. How far this page is really a page about $\mathbb{R}$. It assumes only that $\mathbb{R}$ is a complete ordered field, and it uses completeness solely through the Archimedean property. It is tempting to conclude that every result here holds verbatim for sequences of rationals, and the earlier version of this remark said exactly that. The claim needs splitting in two, because it is true of the statements and not literally true of the proofs.

The statements do transfer. $\mathbb{Q}$ is an ordered field (The rationals form a totally ordered field) and is Archimedean (The rationals are Archimedean), and the absolute value, the triangle inequality and the order arithmetic these proofs run on are established for an arbitrary ordered field, not for $\mathbb{R}$ in particular (Basic properties of the absolute value, The triangle inequality, Ordered field).

The proofs do not transfer by citation, and two places are why. Every convergent sequence is bounded absorbs the first finitely many terms using Every nonempty finite set of reals has a maximum and a minimum, which is stated for finite lists of reals; its proof is an induction that runs in any totally ordered field, but as stated it does not apply to $\mathbb{Q}$. And Every complete ordered field is Archimedean is stated for complete ordered fields, so it says nothing about $\mathbb{Q}$; the $\mathbb{Q}$ counterpart is the separately proved The rationals are Archimedean. A third, more basic point: this library defines convergence and the Cauchy condition only for sequences of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals), so the rational statements are not formulated anywhere here in the first place.

What is therefore claimed, and all that is claimed, is this: rerun any argument on this page over $\mathbb{Q}$, replacing the maximum lemma by the same induction in $\mathbb{Q}$ and Every complete ordered field is Archimedean by The rationals are Archimedean, and it goes through unchanged. That is a statement about the arguments, not a licence to cite the items above with $\mathbb{Q}$ in place of $\mathbb{R}$.