Sequences and Limits

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.

Let $(x_k)$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences), let $x\in \mathbb{R}$, and for $K\in \mathbb{N}$ let $x^{(K)}$ denote the $K$-th tail, $x_j^{(K)}=x_{j+K}$. The following are equivalent (Limits and Cauchy sequences of reals):

1. $(x_k)$ converges to $x$;
2. every tail $x^{(K)}$ converges to $x$;
3. some tail $x^{(K)}$ converges to $x$.

The same three statements with "converges to $x$" replaced by "is Cauchy" are likewise equivalent. Consequently, if two sequences agree from some index on, then either both converge to $x$ or neither does, and either both are Cauchy or neither is.

• This is the lemma that makes the word "eventually" usable. Once it is known that finitely many initial terms are irrelevant, a hypothesis of the form "$P(k)$ for all $k$" can everywhere be weakened to "$P(k)$ eventually" (Sequences of reals: bounded, eventually, frequently, tails, subsequences) at the cost of passing to a tail.

• It also settles the bookkeeping question of where the indexing starts. A text that writes $(x_k{)}_{k\ge 1}$ is describing the first tail of a sequence in the sense used here, and the two have the same limits and the same Cauchy status.

• Boundedness, by contrast, is not a tail property in the same trivial way: a tail of a bounded sequence is bounded, and a sequence with a bounded tail is bounded only because the finitely many omitted terms can be absorbed into the bound, which is exactly the argument of Every convergent sequence is bounded.

Let $(x_k)$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences) and let $x,y\in \mathbb{R}$. If $(x_k)$ converges to $x$ and $(x_k)$ converges to $y$ (Limits and Cauchy sequences of reals), then $x=y$. A sequence therefore has at most one limit, and when a limit exists it may be denoted ${\lim }_k{x}_k$.

• Uniqueness is what licenses the notation ${\lim }_k{x}_k$ and the phrase the limit. Without it the symbol would not denote. This library writes ${\lim }_k{x}_k$ only for sequences already known to converge, exactly as it writes $\sup S$ only for sets already known to have a supremum (Conventions: $\sup \mathrm{\varnothing }$, unbounded sets, and the extended reals).

• The proof uses only that $\mathbb{R}$ is an ordered field in which arbitrarily small positive rationals exist, that is, an Archimedean ordered field (Every complete ordered field is Archimedean). Completeness is not needed: limits are unique in $\mathbb{Q}$ too, where many sequences fail to have one.

• The hypothesis is genuinely about a single sequence having two limits. Two different sequences may of course share a limit, and a sequence with no limit at all is not excluded by anything here.

Let $(x_k)$ be a sequence of reals converging to $x\in \mathbb{R}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals). Then $(x_k)$ is bounded: there is $M\in \mathbb{R}$ with $\mathrm{\mid }{x}_k\mathrm{\mid }\le M$ for every $k\in \mathbb{N}$.

• This is where Every nonempty finite set of reals has a maximum and a minimum does its main work on this page: the first $K$ terms are bounded because a nonempty finite set of reals has a maximum, and that fact is proved once, by induction, rather than re-derived here. It is not the only consumer, and no claim of uniqueness is made: A null sequence times a bounded sequence is null and Conventions for sequences: indexing, eventually, $\lim$, and rational $\epsilon$ cite the same lemma, each for a maximum of finitely many reals.

• The converse is false. A bounded sequence need not converge (FALSE: every bounded sequence converges). What is true in that direction is the Bolzano-Weierstrass theorem, which requires a passage to a subsequence; it is not available at this point in the reading order, being the subject of the next page of this track, Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness, where it is proved.

• Boundedness is exactly what makes A null sequence times a bounded sequence is null applicable to a convergent factor, and that is how boundedness enters the product rule of Algebra of limits: sums, scalar multiples, products and quotients.

Let $(x_k)$ be a sequence of reals converging to $x\in \mathbb{R}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals). Then $(x_k)$ is Cauchy (Limits and Cauchy sequences of reals).

• The converse is a genuine theorem and is not proved here. "Every Cauchy sequence of reals converges" is the completeness of $\mathbb{R}$ in the Cauchy sense. It is the subject of the next page of this track, Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness, where it is proved from the least-upper-bound property, last of the four completeness results there, by way of Bolzano-Weierstrass, which is itself routed through the monotone convergence theorem. That proof is not available at this point in the reading order; the converse itself, for the $\mathbb{R}$ this library constructs, already is, by the different route the next remark records.

• The converse is nevertheless already available in this library, and it should be said plainly rather than left to the next page. The $\mathbb{R}$ used throughout (The real numbers, Sequences of reals: bounded, eventually, frequently, tails, subsequences) is the quotient of the ring of Cauchy sequences of rationals, and The reals are complete proves for precisely that $\mathbb{R}$ that every Cauchy sequence of reals converges to a real. Nothing further is needed to have the converse in hand here; and any other complete ordered field inherits it, since any two are isomorphic by a unique ordered-field isomorphism (Uniqueness of the complete ordered field: $\mathbb{R}$ up to a unique isomorphism). The reason the next page proves it again, from the least-upper-bound property, is that that proof is the form the rest of analysis uses and does not route through a particular construction.

• The splitting $\epsilon \mathrm{/}2+\epsilon \mathrm{/}2$ is the whole content. It is worth noticing that no property of $\mathbb{R}$ beyond the ordered-field axioms and the triangle inequality is used, so the same argument shows that a convergent sequence of rationals is Cauchy in $\mathbb{Q}$.

Let $(x_k)$ and $(y_k)$ be sequences of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences). If $(x_k)$ converges to $0$ (Limits and Cauchy sequences of reals) and $(y_k)$ is bounded, then the product sequence $(x_k{y}_k)$ converges to $0$.

No assumption is made that $(y_k)$ converges. Boundedness alone suffices, and that is why this lemma is stated on its own rather than folded into the product rule for limits.

• The hypothesis on $(y_k)$ looks weaker if it is stated as "eventually bounded", but it is not: a sequence bounded from some index $K$ on is bounded outright. If $\mathrm{\mid }{y}_k\mathrm{\mid }\le M$ for every $k\ge K$, then $M^{\mathrm{\prime }}:=\max \{\mathrm{\mid }{y}_0\mathrm{\mid },\dots ,\mathrm{\mid }{y}_{K-1}\mathrm{\mid },M\}$ 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), and $\mathrm{\mid }{y}_k\mathrm{\mid }\le M^{\mathrm{\prime }}$ for every $k$. That is the same absorption of finitely many initial terms used in Every convergent sequence is bounded.

• Boundedness of $(y_k)$ is essential. Take $x_k:=1\mathrm{/}(k+1)$, which is null (that sequence is shown to converge to $0$ in FALSE: limits preserve strict inequalities), and $y_k:=(k+1{)}^2$. Their product is $x_k{y}_k=k+1$, which is unbounded by the Archimedean property (Every complete ordered field is Archimedean) and hence does not converge at all, since a convergent sequence is bounded (Every convergent sequence is bounded); in particular it is not null. The lemma is therefore sharp in the sense that the bounded factor may not be replaced by an arbitrary one.

• The lemma is the workhorse of the product rule: the identity $x_k{y}_k-xy=x_k(y_k-y)+y(x_k-x)$ exhibits the error as a sum of two products of a null sequence with a bounded one, and boundedness of $(x_k)$ comes from Every convergent sequence is bounded.

Let $(x_k)$ and $(y_k)$ be sequences of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences) converging to $x$ and $y$ respectively (Limits and Cauchy sequences of reals), and let $c\in \mathbb{R}$. Then

$x_k+y_k\to x+y,c{x}_k\to cx,x_k-y_k\to x-y,x_k{y}_k\to xy,$

and if in addition $y\mathrm{\ne }0$ and $y_k\mathrm{\ne }0$ for every $k\in \mathbb{N}$, then

$\frac{1}{y_k}\to \frac{1}{y}\text{and}\frac{x_k}{y_k}\to \frac{x}{y}\mathrm{.}$

The quotient case rests on an eventual lower bound for $\mathrm{\mid }{y}_k\mathrm{\mid }$, proved below rather than assumed: $\mathrm{\mid }{y}_k\mathrm{\mid }>\mathrm{\mid }y\mathrm{\mid }\mathrm{/}2$ for all sufficiently large $k$.