Exercise 2.22

Question

Exercise 22: Show that $\mathbb R^k$ is separable.

ghostofgarborg · Accepted Answer

\def\where{\,|\,}
\def\eps{\varepsilon}

Assume for contradiction that $\mathbb R^k = \bigcup^\infty F_n$ where each $F_n$ is closed and has non-empty interior. 
    Note that $\mathbb Q^k$ is countable. Choose $\epsilon > 0$. 
    If $(x_1, \cdots, x_n) \in \mathbb R^k$, pick rational $r_i$ such that $|r_i - x_i| < \epsilon/ \sqrt k$. 
    Then 
    $$ \|(x_1, \cdots, x_k) - (r_1, \cdots, r_k) \| < \left( \sum_1^k \epsilon^2 / k \right)^{\frac 1 2} = \epsilon $$
    Consequently, any neighborhood around any point of $\mathbb R^k$ contains a point of $\mathbb Q^k$, and $\overline{\mathbb Q^k} = \mathbb R^k$, which implies separability.

Exercise 2.22

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Exercise 2.22

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