I am learning topology by myself:
Let $(X,d)$ be an incomplete metric space. Show that there exists a closed and bounded set $E \subset X$ that is not compact.
My attempt:
Since $X$ is incomplete metric space, then $\exists \{x_n\}$ in $X$ which is a Cauchy sequence but does not converge to any point in $X$.
Since $\{x_n\}$ is a Cauchy squence then let $\epsilon > 0$, $\exists M \in \mathbb{N}$ s.t $\forall n, k \gneq M$, $d(x_n, x_k) < \epsilon$.
Let $B := \sup \{d(x_M, x1), ..., d(x_M, x_{M-1}), \epsilon\}$ then $E := C(x_M, B)$ is a closed and bounded set.
Since $\{x_n\}$ doesn't converge into any point in $X$, so $\forall x \in E \subset X$, $\exists$ bad $\epsilon$ such that $\forall M \in \mathbb{N}$ $\exists n \gneq M$, $d(x, x_n) \gneq \epsilon$
Next, for the sake of contradiction, assume that $E$ is compact.
$$E\subset \bigcup_{x\in E} B(x, \epsilon_x) $$
where $\epsilon_x$ is bad $\epsilon$ for each $x$.
Since $E$ is compact then there exists a finite subcover, let's call it $A = \{a_1, ..., a_k\}$.
So I am stuck here. Could you guys please help me?
If you change the radius of the cover to give you more freedom this should be easier. Assume that the balls are of radius $\frac{1}{2}\epsilon_x$ instead.
Since you obtained a cover, there must be some $a\in A$ such that $\{ x_n \}\cap B(a,\epsilon_a)$ is infinite. I also claim that for any $N\in \mathbb{N}$, there exist $n,m>N$ such that $d(x_n,a)<\frac{1}{2}\epsilon_a$ and $d(x_m,a)>\epsilon_a$. Using the reverse triangle inequality you get a lower bound for $d(x_n,x_m)$. This contradicts the fact that $\{ x_n \}$ is Cauchy.