Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 2, Problem 22

Problem

A metric space is called separable if it contains a countable dense subset. Show that $\mathbb{R}^k$ is separable.

Answer

Consider $\mathbb{Q}^k \subset \mathbb{R}^k$, which is a countable set. Fix $x=(x_1,\cdots,x_k) \in \mathbb{R}^k$ and $\epsilon > 0$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$, there exists $y_i \in \mathbb{Q}$ such that $y_i \in N_{\epsilon/k}(x_i)$ for each $i=1,\cdots,k$. Then, $y=(y_1,\cdots,y_k) \in N_\epsilon(x)$, which implies $\mathbb{Q}^k$ is dense in $\mathbb{R}^k$. 

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