TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-29

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If $(Y, d)$ is complete then $Y^{X}$ is complete in the uniform metric, and so is $C (X, Y)$

Statement

Let $X$ be a nonempty set, let $(Y, d)$ be a complete metric space (Complete metric space: every Cauchy sequence converges in the space) and let $\overset{ˉ}{ρ}$ be the uniform metric on $Y^{X}$ (For a nonempty set $X$ and a metric space $(Y, d)$ the uniform metric $\overset{ˉ}{ρ} (f, g) = \sup_{x} \min {d (f (x), g (x)), 1}$ is a metric on $Y^{X}$ ). Then:

$(Y^{X}, \overset{ˉ}{ρ})$ is a complete metric space.
If in addition $X$ carries a topology, then $C (X, Y)$ with the restriction of $\overset{ˉ}{ρ}$ (Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on $Y^{X}$ and on $C (X, Y)$ ) is a complete metric space.

No choice principle is used. The limit function is defined by a formula, not chosen: a Cauchy sequence in a complete metric space has exactly one limit (A sequence in a metric space has at most one limit), so $x \mapsto \lim_{k} f_{k} (x)$ is a function, and nothing is selected.

Facts & Assumptions

Given: A nonempty set $X$ , a complete metric space $(Y, d)$ , the truncated metric $\overset{ˉ}{d} = \min {d, 1}$ on $Y$ , the uniform metric $\overset{ˉ}{ρ}$ on $Y^{X}$ , and a $\overset{ˉ}{ρ}$ -Cauchy sequence $(f_{k})$ in $Y^{X}$ .

[L1]

$\overset{ˉ}{d} \leq d$ and $\overset{ˉ}{d} \leq 1$ ; if $\overset{ˉ}{d} (u, v) < 1$ then $\overset{ˉ}{d} (u, v) = d (u, v)$ ; and $\overset{ˉ}{d} (u (x), v (x)) \leq \overset{ˉ}{ρ} (u, v)$ for every $x \in X$ , while any real bounding all the values $\overset{ˉ}{d} (u (x), v (x))$ above bounds $\overset{ˉ}{ρ} (u, v)$ ( $\min (d, 1)$ and $d / (1 + d)$ are metrics uniformly equivalent to $d$ , so every metric space carries a bounded metric with the same topology, For a nonempty set $X$ and a metric space $(Y, d)$ the uniform metric $\overset{ˉ}{ρ} (f, g) = \sup_{x} \min {d (f (x), g (x)), 1}$ is a metric on $Y^{X}$ , Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set, Complete ordered field (least-upper-bound property), Suprema and infima are unique).

[L2]

$(x_{k})$ is Cauchy in a metric space when for every real $ε > 0$ there is $K$ with the distance between $x_{m}$ and $x_{n}$ below $ε$ for all $m, n \geq K$ ; the rational and real tests agree (Cauchy sequence in a metric space, The rationals embed densely in the reals).

[L3]

Completeness of $(Y, d)$ : every $d$ -Cauchy sequence in $Y$ converges in $(Y, d)$ , and its limit is unique (Complete metric space: every Cauchy sequence converges in the space, A sequence in a metric space has at most one limit, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ).

[L4]

$x_{k} \to p$ in a metric space means: for every real $ε > 0$ there is $K$ with the distance from $x_{k}$ to $p$ below $ε$ for every $k \geq K$ (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Open ball, closed ball and sphere in a metric space, The rationals embed densely in the reals).

[L5]

The triangle inequality (M3) for $\overset{ˉ}{d}$ ( $\min (d, 1)$ and $d / (1 + d)$ are metrics uniformly equivalent to $d$ , so every metric space carries a bounded metric with the same topology, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[L6]

$C (X, Y)$ is a closed subset of $(Y^{X}, \overset{ˉ}{ρ})$ when $X$ is a nonempty topological space (A uniform limit of continuous functions is continuous, so $C (X, Y)$ is closed in $Y^{X}$ under the uniform metric, claim 3).

[L7]

A closed subset of a complete metric space is complete in the subspace metric (Closed subspaces of complete metric spaces are complete; the converse under countable choice, claim 2, Isometry, isometric embedding, and the subspace metric on a subset).

[L8]

Two elements of $Y^{X}$ are equal exactly when they agree at every point (The topology of pointwise convergence on $Y^{X}$ , which is the product topology, and its restriction to $C (X, Y)$ ).

Proof

technique · direct

1.1givenL1L2choose

Let $x \in X$ and let $ε > 0$ be real; put $η : = \min {ε, 1} / 2$ , a real with $0 < η \leq 1 / 2 < 1$ and $η < ε$ , and take $K$ with $\overset{ˉ}{ρ} (f_{m}, f_{n}) < η$ for all $m, n \geq K$ .

1.2givenL1L2choose

Let $ε > 0$ be real; put $η : = \min {ε, 1} / 4$ , a real with $0 < η \leq 1 / 4 < 1$ and $4 η \leq ε$ , and take $K$ with $\overset{ˉ}{ρ} (f_{m}, f_{n}) < η$ for all $m, n \geq K$ .

2.1step 1.1L1

For $m, n \geq K$ : $\overset{ˉ}{d} (f_{m} (x), f_{n} (x)) \leq \overset{ˉ}{ρ} (f_{m}, f_{n}) < η < 1$ , hence $d (f_{m} (x), f_{n} (x)) = \overset{ˉ}{d} (f_{m} (x), f_{n} (x)) < η < ε$ .

3.1step 2.1L2L3L8

As $ε$ was an arbitrary positive real, step 2.1 makes $(f_{k} (x))$ a $d$ -Cauchy sequence in $Y$ for every $x \in X$ ; by completeness it converges, and its limit is unique, so $f (x) : = \lim_{k} f_{k} (x)$ defines a function $f \in Y^{X}$ with no selection made.

4.1step 3.1step 1.2L1L4L5

Fix $n \geq K$ and $x \in X$ ; since $f_{m} (x) \to f (x)$ in $(Y, d)$ and $\overset{ˉ}{d} \leq d$ , there is $m \geq K$ with $\overset{ˉ}{d} (f_{m} (x), f (x)) < η$ , and then $\overset{ˉ}{d} (f_{n} (x), f (x)) \leq \overset{ˉ}{d} (f_{n} (x), f_{m} (x)) + \overset{ˉ}{d} (f_{m} (x), f (x)) < η + η = 2 η$ .

5.1step 1.2step 4.1L1

Step 4.1 holds for every $x \in X$ , so $2 η$ bounds the values $\overset{ˉ}{d} (f_{n} (x), f (x))$ above and hence $\overset{ˉ}{ρ} (f_{n}, f) \leq 2 η \leq ε / 2 < ε$ , for every $n \geq K$ .

6.1step 5.1L4

As $ε$ was an arbitrary positive real, step 5.1 says $f_{n} \to f$ in $(Y^{X}, \overset{ˉ}{ρ})$ ; so every $\overset{ˉ}{ρ}$ -Cauchy sequence converges in $Y^{X}$ , which is claim 1.

7.1step 6.1L6L7∎

For claim 2, $C (X, Y)$ is closed in the complete space $(Y^{X}, \overset{ˉ}{ρ})$ , so the metric subspace $C (X, Y)$ with the restriction of $\overset{ˉ}{ρ}$ is complete.

Remarks

What completeness of the target buys, pointwise and then uniformly. Step 3.1 produces the limit function pointwise, and that step alone would hold for a merely pointwise Cauchy condition. What the uniform Cauchy condition adds is step 5.1: the same $η$ works at every $x$ , so the bound on $\overset{ˉ}{d} (f_{n} (x), f (x))$ is uniform in $x$ and therefore bounds the supremum.
Step 4.1 chooses nothing. For each fixed $x$ an index $m$ is instantiated and used inside the same sentence; the conclusion $\overset{ˉ}{d} (f_{n} (x), f (x)) < 2 η$ does not mention $m$ , so no function $x \mapsto m$ is ever formed. That is the standard way this library avoids a spurious countable choice.
Completeness is a property of the metric, not of the topology (Complete metric space: every Cauchy sequence converges in the space), and the metric here is $\overset{ˉ}{ρ}$ , built from the truncation $\overset{ˉ}{d} = \min {d, 1}$ . A different metric inducing the same topology on $Y$ need not make $Y$ complete, and then nothing above applies; the hypothesis is that $(Y, d)$ itself is complete.
The classical special case. With $Y = R$ this says that the bounded-or-not real functions on a nonempty set are complete in the uniform metric, and that the continuous ones form a closed, hence complete, subspace. The companion page works $C ([0, 1], R)$ explicitly and compares the uniform metric there with the supremum metric of 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.

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Sources

Uniform convergence (Wikipedia) (standard reference, not scraped)
Complete metric space (Wikipedia) (standard reference, not scraped)
J. Munkres, Topology, 2nd ed., §43 (standard reference, not scraped)

If (Y,d)(Y,d) is complete then YXYX is complete in the uniform metric, and so is C(X,Y)C(X,Y)