Find a metric space X and a subset K of X which is closed and bounded but not compact. 
Find a metric space $X$ and a subset $K$ of $X$ which is closed and bounded but not compact.

I can find a metric space $X$ like the below.

Let $X$ be an infinite set. For $p,q\in X$, define $d(p,q)=\begin{cases}1,&\text{if $p\ne q$}\\0,&\text{if $p=q$}\end{cases}$

Then, with the metric space above, I can find a subset $K$ of $X$, which is a ball which centre is $x$ and radius is $1$.
I know this is closed(since it has no limit points) and bounded.

(I'm confused again... I think $X$ is not an infinite set. Isn't the above triangle the metric space $X$?)
Some helps will be really appreciated!
Thank you! 

HINT: Let $\langle X,d\rangle$ be the metric space that you described in the problem. For any $x\in X$, the open ball $B(x,1)=\{y\in X:d(x,y)<1\}=\{x\}$ is compact: it’s just the singleton set $\{x\}$. However, the closed ball $\overline{B}(x,1)=\{y\in X:d(x,y)\le 1\}$ is very different: $\overline{B}(x,1)=X$ for each each $x\in X$. (Why?) This shows that $X$ itself is a closed, bounded set in $X$
Consider $\{B(x,1):x\in X\}$. This is an open cover of $X$. Does it have a finite subcover? Is $X$ compact?
