91Ó°ÊÓ

Let \(K\) be a sequentially compact subset of \(\mathbb{R}^{n}\) and suppose that \(\mathcal{O}\) is an open subset of \(\mathbb{R}^{n}\) that contains \(K\). Prove that there is some positive number \(r\) such that for any point \(\mathbf{u}\) in \(K, \mathcal{B}_{r}(\mathbf{u}) \subseteq \mathcal{O}\)

Let \(A\) be a subset of \(\mathbb{R}^{n}\) and let the function \(f: A \rightarrow \mathbb{R}\) be continuous. a. If \(A\) is bounded, is \(f(A)\) bounded? b. If \(A\) is closed, is \(f(A)\) closed?