1. Analysis - Compactness

Hello!

I am trying to prove that K is compact (from the definition of compactness), where $\displaystyle K \subset R$ is defined by $\displaystyle K = \{\frac{1}{n} : n \in N\} \cup \{0\}$.

Any help is greatly appriciated!
Thanks!
Matt

2. Originally Posted by matt.qmar
Hello!

I am trying to prove that K is compact (from the definition of compactness), where $\displaystyle K \subset R$ is defined by $\displaystyle K = \{\frac{1}{n} : n \in N\} \cup \{0\}$
If $\displaystyle \varepsilon > 0$ then the open interval $\displaystyle \left( {0,\varepsilon } \right)$ contain almost all the points of $\displaystyle K$.

3. Originally Posted by matt.qmar
Hello!

I am trying to prove that K is compact (from the definition of compactness), where $\displaystyle K \subset R$ is defined by $\displaystyle K = \{\frac{1}{n} : n \in N\} \cup \{0\}$.

Any help is greatly appriciated!
Thanks!
Matt
It may be more clear (at least to me) that in any metric space, if $\displaystyle x_n\to x$ then $\displaystyle \{x\}\cup\{x_n:n\in\mathbb{N}\}$ is compact. Use Plato's idea of looking at a neighborhood of the convergence point.