91Ó°ÊÓ

Let \(\left(x_{n}\right)_{n \geq 0}\) be a sequence in \(\mathbb{R}^{p} . a \in \mathbb{R}^{p}\) is called an accumulation value of the sequence \(\left(x_{n}\right)\) if for each \(\varepsilon\)-disk \(U_{\varepsilon}(a)\) there are infinitely many indices \(n\) such that \(x_{n} \in U_{\varepsilon}(a)\). Show (BOLZANO-WEIERSTRASS Theorem): Any bounded sequence \(\left(x_{n}\right), x_{n} \in\) \(\mathbb{R}^{p}\) has an accumulation point. A subset \(K \subseteq \mathbb{R}^{p}\) is called sequentially compact if each sequence \(\left(x_{n}\right)_{n \geq 0}\) with \(x_{n} \in K\) has (at least) one accumulation point in \(K\) Show: For a subset \(K \subseteq \mathbb{R}^{p}\) the following are equivalent: (a) \(K\) is compact, (b) \(K\) is sequentially compact. Remark. These equivalences hold for any metric space.