I'm stuck on how to show one of these is compact, and I want to verify my method for the other.
This one I am stuck on:
Proposition Let $(\mathbb{Q},d)$ be a metric space with $d(a,b)=|a-b|$. Let $E \subset \mathbb{R}$ and let $E=\{r \in \mathbb{Q}: r^2 < 2\}$. Show $E$ is not compact.
This one I believe I have solved and want to verify:
Proposition Let $\{x_n\}$ be a convergent sequence in $\mathbb{R}$ that converges to $x$. Let $E=\{x_n : n=1,2,3...\}\cup \{x\}$. Then $E$ is compact.
I believe I demonstrated this by the following. Since $\{x_n\}$ is convergent fix $\epsilon =1$ and then for $N \in \mathbb{N}$, there exists an $n \geq N$ such that $|x_n-x|<1$. Then take $\bar{\epsilon}=\max\{d(x_1,x)...d(x_N,x), 1\}$. Clearly $E \subset (\bigcup\limits_{k=1}^{N} N(x_k, \bar{\epsilon}))\bigcup N(x, 1))$ - this is a finite subcover of $N+1$ sets.