ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-26

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The supremum metric $d_{\infty} (f, g) = \sup_{x} ∣ f (x) - g (x) ∣$ on the bounded real-valued functions on a set

Example

Let $S$ be a nonempty set, let $B (S)$ be the set of bounded functions $S \to R$ and let $d_{\infty} (f, g) = \sup {∣ f (s) - g (s) ∣ : s \in S}$ be the supremum metric; that this is a metric is The supremum metric $d_{\infty} (f, g) = \sup_{x} ∣ f (x) - g (x) ∣$ is a metric on the bounded real-valued functions on a nonempty set and is quoted here rather than reproved. This example records two things about it.

The constants form an isometric copy of the real line. For $c \in R$ let $\hat{c} \in B (S)$ be the constant function with value $c$ . Then $c \mapsto \hat{c}$ is an isometric embedding of $(R, d_{R})$ into $(B (S), d_{\infty})$ (Isometry, isometric embedding, and the subspace metric on a subset, The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded): $d_{\infty} (\hat{c}, \hat{c}^{'}) = ∣ c - c^{'} ∣ (c, c^{'} \in R) .$
The supremum need not be attained. Take $S = N$ , so that $B (N)$ is the set of bounded sequences of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences), and put $f (k) : = 1 - \frac{1}{k + 1} (k \in N), g : = \hat{0} .$ Then $f, g \in B (N)$ , $d_{\infty} (f, g) = 1$ , and $∣ f (k) - g (k) ∣ < 1$ for every $k \in N$ . So $d_{\infty} (f, g)$ is a supremum that is not a maximum, and no single point of $S$ realises the distance.

The index shift in $1 / (k + 1)$ is forced: $N$ contains $0$ (The natural numbers $N$ (von Neumann)) and sequences here are indexed from $0$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences), so $1 / k$ would be undefined at $k = 0$ .

Facts & Assumptions

Given: A nonempty set $S$ ; reals $c, c^{'}$ ; the constant functions $\hat{c}, {\hat{c}}^{'} : S \to R$ ; and, for $S = N$ , the functions $f (k) = 1 - 1 / (k + 1)$ and $g = \hat{0}$ , together with $D (f, g) : = {∣ f (k) - g (k) ∣ : k \in N}$ .

[L1]

The supremum metric is a metric on $B (S)$ for nonempty $S$ , and $d_{\infty} (u, v)$ is the least upper bound of ${∣ u (s) - v (s) ∣ : s \in S}$ (The supremum metric $d_{\infty} (f, g) = \sup_{x} ∣ f (x) - g (x) ∣$ is a metric on the bounded real-valued functions on a nonempty set, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[L2]

Least upper bounds: a nonempty subset of $R$ bounded above has a unique least upper bound (Complete ordered field (least-upper-bound property), Suprema and infima are unique); bounded subsets of $R$ are as in Lower bound, bounded below, bounded set.

[L3]

Epsilon characterisation of the supremum: for a nonempty $T \subseteq R$ bounded above and an upper bound $u$ of $T$ , one has $u = \sup T$ if and only if for every real $ε > 0$ there is $t \in T$ with $u - ε < t$ (Epsilon characterisation of the supremum).

[L4]

Reciprocal Archimedean property: for every real $ε > 0$ there is a natural $n \geq 1$ with $1 / n < ε$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ , Every complete ordered field is Archimedean); and $m \cdot 1_{R} > 0$ with $1 / m > 0$ for $m \geq 1$ (Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order).

[L5]

Absolute value: $∣ t ∣ = t$ for $t \geq 0$ , $∣ t ∣ \geq 0$ (Basic properties of the absolute value, Absolute value in an ordered field); and the usual metric of $R$ is $d_{R} (c, c^{'}) = ∣ c - c^{'} ∣$ (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded).

[L6]

Isometric embedding: a map preserving distances exactly (Isometry, isometric embedding, and the subspace metric on a subset); trichotomy and transitivity of the order of $R$ (Ordered field, Complete ordered field (least-upper-bound property)).

Verification

technique · direct

1.1

Each constant function $\hat{c}$ has range ${c}$ , a bounded subset of $R$ , so $\hat{c} \in B (S)$ ; and ${∣ \hat{c} (s) - {\hat{c}}^{'} (s) ∣ : s \in S} = {∣ c - c^{'} ∣}$ , a nonempty one-element set whose least upper bound is $∣ c - c^{'} ∣$ itself.

givenL1L2L5

2.1

For $S = N$ : each $k + 1$ is a natural $\geq 1$ , so $1 / (k + 1)$ is a positive real and $0 \leq f (k) = 1 - 1 / (k + 1) < 1$ ; hence the range of $f$ is bounded and $f \in B (N)$ , while $g = \hat{0}$ is bounded by step 1.1, and $D (f, g) = {1 - 1 / (k + 1) : k \in N}$ is nonempty with $1$ as an upper bound.

step 1.1L4L5

2.2

Claim 1: by step 1.1, $d_{\infty} (\hat{c}, {\hat{c}}^{'}) = ∣ c - c^{'} ∣ = d_{R} (c, c^{'})$ for all reals $c, c^{'}$ , which is exactly the statement that $c \mapsto \hat{c}$ is an isometric embedding of $(R, d_{R})$ into $(B (S), d_{\infty})$ .

step 1.1L1L5L6

3.1

$d_{\infty} (f, g) = 1$ : the number $1$ is an upper bound of $D (f, g)$ by step 2.1, and for an arbitrary real $ε > 0$ choose a natural $n \geq 1$ with $1 / n < ε$ and put $k : = n - 1$ , a natural since $n \geq 1$ , so that $1 - 1 / (k + 1) = 1 - 1 / n > 1 - ε$ ; by the epsilon characterisation $1$ is the least upper bound.

step 2.1L3L4L6

4.1

The supremum is not attained: every element of $D (f, g)$ has the form $1 - 1 / (k + 1)$ with $1 / (k + 1) > 0$ , hence is $< 1 = d_{\infty} (f, g)$ , so no $k \in N$ satisfies $∣ f (k) - g (k) ∣ = d_{\infty} (f, g)$ .

step 2.1step 3.1L4L6

5.1

Claims 1 and 2 are established, by step 2.2 and by steps 3.1 and 4.1 respectively.

step 2.2step 3.1step 4.1∎

Remarks

Why the constants matter. The isometric copy of $R$ inside $B (S)$ shows that $(B (S), d_{\infty})$ is unbounded whenever $S \neq \emptyset$ , since $R$ is (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded, An isometric embedding is injective and carries the metric topology of the source onto the subspace topology of its image).
With $S = N$ this is the space usually written $ℓ^{\infty}$ , the bounded real sequences with the supremum metric. Its completeness and its separability are questions for later pages and are not touched here.
Attainment is a genuinely different question from existence. The supremum exists because the set is nonempty and bounded above (Complete ordered field (least-upper-bound property)); whether it lies in the set is exactly the question of whether the sup is a maximum, and claim 2 answers it negatively for a specific pair.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

42 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

Uniform norm (Wikipedia) (standard reference, not scraped)
Sequence space (Wikipedia) (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 7 (standard reference, not scraped)
Isometry (Wikipedia) (standard reference, not scraped)

The supremum metric d∞(f,g)=sup⁡x∣f(x)−g(x)∣d∞​(f,g)=supx​∣f(x)−g(x)∣ on the bounded real-valued functions on a set