Is the following a correct way of showing that the set $(0,1]$ is not compact or I am missing something.
We will prove that it does not satisfy the Bolzano Weierstrass Theorem, i.e., there exists a sequence in $E=(0,1]$ with a converging subsequence that does not have a limit in $(0,1].$\
Consider the sequence $\{a_n\}$ defined by $\{\frac{1}{n} \}_{n=1}^{\infty}$i.e., we have $(1, \frac{1}{2}, \frac{1}{3}, \dots).$ Clearly $\{a_n\} \subset (0,1]$ and it is bounded by 1. Then, if we take it as a subsequence of itself, we see that its limit is $0$ which is not in $(0,1]$ hence $(0,1]$ is not compact as required.