= Cauchy sequence
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{title2=$\forall\varepsilon>0\;\exists N\;\forall m,n\geq N:\ d(x_m,x_n)<\varepsilon$}
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= Cauchy
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A <Cauchy sequence> is a <sequence> whose sufficiently late terms are all close to one another. In a <metric space> $(X,d)$, the precise condition is
$$
\forall\varepsilon>0\;\exists N\in\mathbb N_{>0}\;\forall m,n\geq N:\ d(x_m,x_n)<\varepsilon.
$$
For <real numbers> with their usual <metric>, $d(x_m,x_n)=|x_m-x_n|$. The useful feature is that the definition compares computed terms to other computed terms: it does not require an already known <limit of a sequence>. The one index $N$ must control every pair of later indices, not merely successive terms. https://www.jirka.org/ra/html/sec_cauchy.html[Lebl, Basic Analysis I, §2.4: Cauchy sequences and the role of all late pairs]{ref}.
For an explicit example, take $q_n=1-2^{-n}$ for $n\geq1$. Its <tail of a sequence> beginning at $N$ lies in $[1-2^{-N},1)$, and for every $m,n\geq N$,
$$
|q_m-q_n|=|2^{-n}-2^{-m}|<2^{-N}.
$$
Choose $N$ with $2^{-N}<\varepsilon$ to certify the entire future <sequence>. For $\varepsilon=0.1$, $N=4$ works because $2^{-4}=0.0625$. The <limit of a sequence> is one, but the pairwise bound establishes the <Cauchy sequence> property without first needing that answer. By contrast, $a_n=(-1)^n$ is not a <Cauchy sequence>: arbitrarily late <even integers> and <odd integers> give values separated by two. https://www.ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_lec102.pdf[MIT 18.100A, Lecture 10, Definition 1, Negation 3 and Example 4]{ref}.
\Image[/cauchy-tail-clustering-versus-alternation.png]
{title=A <Cauchy sequence> confines every sufficiently late pair to a small band; persistent alternation cannot do so}
{description=The original geometric example $q_n=1-2^{-n}$ has all terms with $n\geq4$ in $[15/16,1)$, a band narrower than $0.1$. Only finitely many terms are drawn; the formula certifies the entire tail. The alternating example $(-1)^n$ continues to attain both $-1$ and $1$ arbitrarily late.}
Tiny successive changes do not provide this guarantee. For the <harmonic numbers> $H_n=\sum_{j=1}^n1/j$, the successive difference $H_{n+1}-H_n=1/(n+1)$ tends to zero, but
$$
H_{2n}-H_n=\sum_{j=n+1}^{2n}\frac1j\geq n\frac1{2n}=\frac12.
$$
Thus, beyond every proposed $N$, choosing $n\geq N$ and $m=2n$ defeats the fixed tolerance $\varepsilon=1/2$. Even every fixed positive lag $h$ passes the weaker test: $0<H_{n+h}-H_n\leq h/(n+1)\to0$. The failing lag grows with $n$, as explained more generally by <convergence from arbitrary variable-lag increments>. https://www.jirka.org/ra/html/sec_cauchy.html[Lebl, Basic Analysis I, §2.4: shrinking fixed-lag increments do not imply the Cauchy condition]{ref}.
\Image[/harmonic-successive-versus-doubling-gaps.png]
{title=<Harmonic numbers> take vanishing successive steps while retaining a nonvanishing gap between indices $n$ and $2n$}
{description=The blue curve is $H_{n+1}-H_n=1/(n+1)$; the red curve is $H_{2n}-H_n$. The exact lower bound $1/2$ holds for every positive <integer> $n$, not just the plotted sample. The shaded region between $n$ and $2n$ in the right panel illustrates how many individually small terms produce a substantial gap.}
Every <convergent sequence> in a <metric space> is a <Cauchy sequence>: apply the <triangle inequality> to two late terms and their common <limit of a sequence>. The reverse implication requires <completeness>. It holds in the <real numbers> by <completeness of the real numbers>, but can fail in the <rational numbers>: the rational endpoints from the <bisection method> approaching $\sqrt2$ are a <Cauchy sequence> with no rational <limit of a sequence>. The existing construction in <completeness of the real numbers> shows how this happens. A <Cauchy sequence> is always a <bounded sequence>; in $\mathbb R$, boundedness gives a <convergent subsequence> by the <Bolzano-Weierstrass theorem>, and the pairwise condition forces the whole <sequence> to share its limit. https://www.ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_lec102.pdf[MIT 18.100A, Lecture 10, Theorems 5–7: boundedness, subsequences and real completeness]{ref}.
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