Exercise 2.12

Question

Exercise 12: 
    Let $K\subset\mathbf R$ consist of 0 and the numbers $1/n$, for $n=1,2,3,\ldots$. Prove that $K$ is compact directly from the definition
    (without using the Heine-Borel theorem).

ghostofgarborg · Accepted Answer

\def\eps{\varepsilon}
\def\ints{\mathbb{Z}}

Let $\mathcal U$ be an open cover for $K$. Then there exists a $ U \in \mathcal U$ 
    for which $0 \in U$. By openness, there exists an $\epsilon > 0$ such that 
    $N_\epsilon(0) \subset U$. By the archimedean property, there exists an $N$ 
    such that $N\epsilon > 1$, i.e. such that $\frac 1 n < \epsilon$ for $n > N$.
    Therefore, $N_\epsilon(0)$ contains all but a finite number of elements of $K$.
    We can now pick a set in $\mathcal U$ for each of the remaining points, and obtain a finite cover.

Exercise 2.12

Answers

Comments

Exercise 2.12

Answers

Comments

Add answer