LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A pseudocompact subset of $\mathbb{R}^n$ is bounded

Statement

Let $n\ge1$ . Every pseudocompact subset $A\subseteq\mathbb{R}^n$ is bounded for the Euclidean metric.

Facts & Assumptions

Given: A pseudocompact subset $A\subseteq\mathbb{R}^n$ , where $\mathbb{R}^n$ has the Euclidean metric $d_2$ .

[L1]

The Euclidean norm $x\mapsto\lVert x\rVert_2$ is continuous from $(\mathbb{R}^n,d_2)$ to $\mathbb{R}$ (The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$ ).

[L2]

A pseudocompact space has bounded image under every continuous real-valued map (Pseudocompact space: every continuous real-valued function has bounded image).

[L3]

A subset of a metric space is bounded when it is empty or lies in some open ball (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space); a bounded set of reals has an upper bound (Lower bound, bounded below, bounded set).

Proof

technique · direct

1.1

The restriction $N:A\to\mathbb{R}$ , $N(x)=\lVert x\rVert_2$ , is continuous, because the ambient norm is continuous by [L1] and $A$ has the subspace topology.

1.2

Pseudocompactness gives that $N[A]$ is bounded. If $A\ne\varnothing$ , choose an upper bound $M$ of $N[A]$ ; then $M\ge0$ because every norm is nonnegative.

L2L3choose

2.1

If $A=\varnothing$ it is bounded. Otherwise every $x\in A$ satisfies $d_2(x,0)=\lVert x\rVert_2\le M<M+1$ , so $A\subseteq B_{d_2}(0,M+1)$ .

step 1.2L3

3.1

Thus $A$ is bounded in both cases.

step 2.1∎

Depends on

Used by

For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 121 results over 23 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

Pseudocompact space (standard reference, not scraped)
Heine-Borel theorem (Wikipedia) (standard reference, not scraped)

A pseudocompact subset of Rn\mathbb{R}^nRn is bounded