I am referring to Exercise 7.4.17 from the book Basic Analysis I by Lebl. I was only able to find this, but it is a proof verification using a somewhat different approach.
Let $(X,d)$ be an incomplete metric space. Show that there exists a closed and bounded set $E\subset X$ that is not compact.
I know that incompleteness implies there is some $(x_n)\subset X$ where $(x_n)$ is Cauchy and it does not converge anywhere in $X$. Furthermore, any subsequence of $(x_n)$ does not converge anywhere in $X$. I was thinking of letting $E$ be formed by all of the sequence itself or just some subsequence of $(x_n)$ knowing that $E$ will be bounded. I am just not sure how to proceed showing that this E would be closed and not compact.
Any help or hints are appreciated.