Lebl-Basic-Analysis-I · Problem 7.6.1

Metric Spaces

Mathematics · fixed-point-theorem

Problem

Let JJ be a closed and bounded interval and Y≔{f∈C([−h,h],R):f([−h,h])⊂J}Y \coloneqq \bigl\{ f \in C\bigl([-h,h],\mathbb{R}\bigr) : f\bigl([-h,h]\bigr) \subset J \bigr\}. Show that Y⊂C([−h,h],R)Y \subset C\bigl([-h,h],\mathbb{R}\bigr) is closed. Hint: JJ is closed.

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