Let K be a nonempty subset of R^n, where n 1. Which of the following statements must be true? I. If K is compact, then every continuous real-valued function defined on K is bounded. II. If every continuous real-valued function defined on K is bounded, then K is compact. III. If K is compact, then K is connected.

Question

Let K be a nonempty subset of R^n, where n > 1. Which of the following statements must be true? I. If K is compact, then every continuous real-valued function defined on K is bounded. II. If every continuous real-valued function defined on K is bounded, then K is compact. III. If K is compact, then K is connected.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the truth of the statements, we must use the definitions of compactness, boundedness, and connectedness in $ \mathbb{R}^n $. ‖ ‖ step 2 ⋮ Statement I: If $ K $ is compact, then every continuous real-valued function defined on $ K $ is bounded. This is true by the Extreme Value Theorem. ‖ ‖ step 3 ⋮ Statement II: If every continuous real-valued function defined on $ K $ is bounded, then $ K $ is compact. This is not necessarily true; boundedness alone does not imply compactness. ‖ ‖ step 4 ⋮ Statement III: If $ K $ is compact, then $ K $ is connected. This is not necessarily true; compactness does not imply connectedness. ‖ ∻Answer∻ ⚹ Statement I is true; Statements II and III are false. ⚹ ∻Key Concept∻ ⚹Compactness and Continuous Functions⚹ ∻Explanation∻ ⚹A compact subset of $ \mathbb{R}^n $ ensures that every continuous function defined on it is bounded (Extreme Value Theorem), but compactness does not imply connectedness, and boundedness of functions alone does not imply compactness of the set.⚹◊What is the relationship between compactness and boundedness of a nonempty subset K in $\mathbb{R}^n$?⍭ Generate me a similar question◊

Sia