LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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Every Cauchy sequence in a metric space is bounded

Statement

Let $(X, d)$ be a metric space (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric) and let $(x_{k})$ be a Cauchy sequence in $X$ (Cauchy sequence in a metric space). Then its range $R : = {x_{k} : k \in N}$ is a bounded subset of $X$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space): there are a point $x_{0} \in X$ and a real $r > 0$ with $R \subseteq B (x_{0}, r)$ (Open ball, closed ball and sphere in a metric space).

Consequently $R$ is nonempty and bounded, so $diam (R)$ exists (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Facts & Assumptions

Given: A metric space $(X, d)$ and a Cauchy sequence $(x_{k})$ in $X$ ; write $R : = {x_{k} : k \in N}$ .

[A1]

Cauchyness at the real value $1$ : there is $N \in N$ with $d (x_{m}, x_{n}) < 1$ for all $m, n \geq N$ (Cauchy sequence in a metric space, The rationals embed densely in the reals).

[L1]

A nonempty finite set of reals has a maximum, and every element of the set is at most that maximum (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set).

[L2]

A metric takes nonnegative values (Nonnegativity of a metric is a consequence of the other axioms, not an axiom).

[L3]

Membership in a ball: $y \in B (x, r)$ means $d (x, y) < r$ , and the radius is a positive real (Open ball, closed ball and sphere in a metric space).

[L4]

A subset $A \subseteq X$ is bounded when $A = \emptyset$ or $A \subseteq B (x_{0}, r)$ for some $x_{0} \in X$ and real $r > 0$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Proof

technique · direct

1.1

Fix $N$ as in [A1], so that $d (x_{m}, x_{n}) < 1$ whenever $m, n \geq N$ ; in particular $d (x_{k}, x_{N}) < 1$ for every $k \geq N$ .

1.2

The set $S : = {d (x_{j}, x_{N}) : j \leq N} \cup {1}$ is a nonempty finite set of reals, so it has a maximum $M : = \max S$ , and $M \geq 1 > 0$ .

L1L2

2.1

Put $r : = M + 1$ , a real with $r > 1 > 0$ . For $j \leq N$ we have $d (x_{j}, x_{N}) \leq M < r$ , and for $k \geq N$ we have $d (x_{k}, x_{N}) < 1 \leq M < r$ ; every index is of one of the two kinds, so $d (x_{k}, x_{N}) < r$ for every $k \in N$ .

step 1.1step 1.2choose

3.1

Hence $x_{k} \in B (x_{N}, r)$ for every $k$ , that is $R \subseteq B (x_{N}, r)$ with $x_{N} \in X$ and $r > 0$ , so $R$ is bounded; and $R$ is nonempty because it contains $x_{0}$ .

step 2.1L3L4∎

Remarks

The maximum is taken over $j \leq N$ and not over $j < N$ , and the extra element $1$ is in the set as well. Both are deliberate. Indices run from $0$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences), so $N = 0$ is possible and ${d (x_{j}, x_{N}) : j < N}$ would then be empty, which has no maximum (Maximum and minimum of a set); adjoining $1$ makes the set nonempty in every case and simultaneously covers the tail bound.
Boundedness of the range is strictly weaker than Cauchyness. The sequence $x_{k} = (- 1)^{k}$ in $R$ has bounded range and is not Cauchy, since two consecutive terms are always at distance $2$ . So this lemma cannot be reversed, and nothing on this page reverses it.
What the lemma is for. It is what makes $diam$ available for the tails of a Cauchy sequence, which is the form in which Cauchyness enters the converse half of In a complete metric space nested nonempty closed sets whose diameters tend to $0$ meet in exactly one point, and this property characterises completeness.

Depends on

Used by

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Cauchy sequence (Wikipedia) (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (standard reference, not scraped)