ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The metrics $d_1$ , $d_2$ and $d_\infty$ on $\mathbb{R}^n$ are metrics and are Lipschitz equivalent, with explicit constants

Example

Let $n \ge 1$ be a natural number and let $\mathbb{R}^n$ carry the three metrics

$d_1(x,y) = \sum_{k<n}|x_k-y_k|, \qquad d_2(x,y) = \sqrt{\textstyle\sum_{k<n}(x_k-y_k)^2}, \qquad d_\infty(x,y) = \max\{|x_k-y_k| : k<n\}$

of $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$ , and $d_1$ , $d_2$ , $d_\infty$ are metrics on it, where $\mathbb{R}^n$ is the set of functions from the von Neumann natural $n$ to $\mathbb{R}$ . All three are metrics (Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric); that is $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$ , and $d_1$ , $d_2$ , $d_\infty$ are metrics on it and is quoted here rather than reproved. What this example adds is that the three are Lipschitz equivalent with explicit constants (Topologically, uniformly and Lipschitz equivalent metrics on a set): for all $x, y \in \mathbb{R}^n$ ,

$d_\infty(x,y) \;\le\; d_2(x,y) \;\le\; \sqrt{n}\; d_\infty(x,y), \qquad d_\infty(x,y) \;\le\; d_1(x,y) \;\le\; n\; d_\infty(x,y), \qquad d_2(x,y) \;\le\; d_1(x,y) \;\le\; \sqrt{n}\; d_2(x,y).$

Consequently the three are uniformly equivalent and topologically equivalent (Lipschitz equivalence implies uniform equivalence implies topological equivalence), so they determine the same open sets, the same convergent sequences and the same continuous maps on $\mathbb{R}^n$ .

The constants are best possible: taking $x - y$ with a single nonzero coordinate gives equality in $d_\infty = d_2 = d_1$ , and taking all $n$ coordinates equal in absolute value gives $d_1 = n\, d_\infty$ and $d_2 = \sqrt{n}\, d_\infty$ . Those two remarks are not needed for the equivalence and are not proved below.

Facts & Assumptions

Given: A natural $n \ge 1$ , elements $x, y \in \mathbb{R}^n$ , the list $u_k := x_k - y_k$ for $k < n$ , and the abbreviations $M := d_\infty(x,y)$ , $S := d_1(x,y) = \sum_{k<n}|u_k|$ and $Q := \sum_{k<n} u_k^2$ , so that $d_2(x,y) = \sqrt{Q}$ ; the canonical natural $n$ is here read inside $\mathbb{R}$ as $n \cdot 1_{\mathbb{R}}$ .

[L1]

$d_1$ , $d_2$ and $d_\infty$ are metrics on $\mathbb{R}^n$ ( $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$ , and $d_1$ , $d_2$ , $d_\infty$ are metrics on it, Metric space: $d(x,y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, Finite sums and finite products, by recursion).

[L2]

Laws of finite sums (Laws of finite sums and finite products, Finite sums and finite products, by recursion): monotonicity, scaling, $\sum_{k<n}\lambda = n\lambda$ , a sum of nonnegative terms is nonnegative, and each single term is at most such a sum.

[L3]

The maximum of a nonempty finite set of reals exists, is one of its elements and bounds the set above (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set).

[L4]

Square roots (Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$ ; the positives are $\{x^2 : x \neq 0\}$ ): every $c \ge 0$ has a unique $\sqrt{c} \ge 0$ with $(\sqrt{c})^2 = c$ ; hence $\sqrt{t^2} = t$ for $t \ge 0$ and $\sqrt{ab} = \sqrt{a}\sqrt{b}$ for $a,b \ge 0$ , both by uniqueness. Squaring is monotone on the nonnegatives, $s \le t \iff s^2 \le t^2$ (Squaring is monotone on the nonnegatives), so the same holds for square roots.

[L5]

Cauchy-Schwarz in root form (The Cauchy-Schwarz inequality for finite sums): $\big|\sum_{k<n} a_kb_k\big| \le \sqrt{\sum_{k<n}a_k^2}\,\sqrt{\sum_{k<n}b_k^2}$ .

[L6]

Absolute value (Basic properties of the absolute value, Absolute value in an ordered field, Integer powers $a^m$ ): $|t| \ge 0$ , $|t|^2 = t^2$ , and $|t| = t$ for $t \ge 0$ .

[L7]

Order arithmetic: multiplying an inequality by a nonnegative element preserves it and inequalities may be added, in the strict forms of Sign rules for products and monotonicity of multiplication and Order is preserved by adding a constant and by adding inequalities together with the case of equality settled by totality (Ordered field, Complete ordered field (least-upper-bound property)); and $n \cdot 1_{\mathbb{R}} > 0$ for $n \ge 1$ (Canonical naturals are positive and strictly increasing).

[L8]

Lipschitz equivalence and the hierarchy (Topologically, uniformly and Lipschitz equivalent metrics on a set, Lipschitz equivalence implies uniform equivalence implies topological equivalence).

Verification

technique · direct

1.1

Since $n \ge 1$ the set $\{|u_k| : k<n\}$ is nonempty and finite, so $M$ exists, equals $|u_j|$ for some $j < n$ , satisfies $M \ge 0$ , and bounds every $|u_k|$ above.

L3L6

1.2

The reals $n \cdot 1_{\mathbb{R}}$ , $\sqrt{n}$ , $Q$ and $S$ are all nonnegative, and $n \cdot 1_{\mathbb{R}} > 0$ , so $\sqrt{n} > 0$ ; also $M^2 = u_j^2$ and $u_k^2 = |u_k|^2$ for every $k$ .

L2L4L6L7

2.1

First chain: $M^2 = u_j^2 \le Q$ because a single nonnegative term is at most the sum, so $M = \sqrt{M^2} \le \sqrt{Q} = d_2(x,y)$ ; and $Q = \sum_{k<n}u_k^2 \le \sum_{k<n}M^2 = n M^2$ by monotonicity and scaling, so $d_2(x,y) = \sqrt{Q} \le \sqrt{nM^2} = \sqrt{n}\,M$ .

step 1.1step 1.2L2L4L7

2.2

Second chain: $M = |u_j| \le S$ because a single nonnegative term is at most the sum; and $S = \sum_{k<n}|u_k| \le \sum_{k<n}M = nM$ by monotonicity and scaling.

step 1.1step 1.2L2L7

2.3

Third chain: for every $k < n$ , multiplying $|u_k| \le S$ by the nonnegative $|u_k|$ gives $u_k^2 = |u_k|^2 \le |u_k| S$ , so summing and scaling gives $Q \le S \sum_{k<n}|u_k| = S^2$ and hence $d_2(x,y) = \sqrt{Q} \le \sqrt{S^2} = S = d_1(x,y)$ ; and Cauchy-Schwarz applied to the lists $a_k := |u_k|$ and $b_k := 1$ gives $S = \big|\sum_{k<n}|u_k|\cdot 1\big| \le \sqrt{\sum_{k<n}u_k^2}\ \sqrt{\sum_{k<n}1} = \sqrt{Q}\,\sqrt{n} = \sqrt{n}\, d_2(x,y)$ .

step 1.1step 1.2L2L4L5L6L7

3.1

The three chains are exactly Lipschitz equivalences with positive constants: $1 \cdot d_\infty \le d_2 \le \sqrt{n}\, d_\infty$ , $1 \cdot d_\infty \le d_1 \le n\, d_\infty$ and $1 \cdot d_2 \le d_1 \le \sqrt{n}\, d_2$ , the constants $1$ , $\sqrt{n}$ and $n$ all being positive.

step 2.1step 2.2step 2.3L7L8

4.1

Hence any two of $d_1$ , $d_2$ , $d_\infty$ are Lipschitz equivalent, and therefore uniformly equivalent and topologically equivalent; all three induce the same topology on $\mathbb{R}^n$ .

step 3.1L1L8∎

Remarks

The constants blow up with the dimension, and that is the whole point of the distinction. The comparison $d_1 \le n\, d_\infty$ is Lipschitz for each fixed $n$ and useless uniformly in $n$ , so no pair of constants serves all dimensions at once. Whether an analogue survives on spaces of infinite sequences is a question for a later page and is not addressed here.
Only $n \ge 1$ is treated, because $d_\infty$ is a maximum over the index set $\{0,\dots,n-1\}$ and that set is empty when $n = 0$ ( $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$ , and $d_1$ , $d_2$ , $d_\infty$ are metrics on it). For $n = 0$ the space $\mathbb{R}^0$ is a single point and $d_1$ , $d_2$ are identically $0$ on it, while $d_\infty$ is not defined there at all, so there is nothing to compare.
Minkowski is not used here. The triangle inequalities were settled in $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$ , and $d_1$ , $d_2$ , $d_\infty$ are metrics on it; what this page needs is only the comparison of the three values, and that runs on the finite-sum laws and Cauchy-Schwarz.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 87 results over 30 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Lp space (Wikipedia) (standard reference, not scraped)
Equivalence of metrics (Wikipedia) (standard reference, not scraped)
Taxicab geometry (Wikipedia) (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)
R. Gardner, Introduction to Topology, notes on Munkres Section 20: The Metric Topology (East Tennessee State University) (standard reference, not scraped)

The metrics d1d_1d1​, d2d_2d2​ and d∞d_\inftyd∞​ on Rn\mathbb{R}^nRn are metrics and are Lipschitz equivalent, with explicit constants