Basic Analysis I — Real Analysis · Problem 7.6.1

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · fixed-point-theorem

Problem

Let

J

be a closed and bounded interval and

Y \coloneqq \bigl\{f \in C\bigl([-h,h],\mathbb{R}\bigr) : f\bigl([-h,h]\bigr) \subset J\bigr\}

. Show that

Y \subset C\bigl([-h,h],\mathbb{R}\bigr)

is closed. Hint:

J

is closed.