Why are locally compact groups Weil complete?
Note: A topological group $G$ is Weil complete if every left Cauchy net in $G$ is convergent.
Thank you, and sorry if I have bad writing.
Take a left Cauchy net $(u_i)$. Choose $V$ a compact neighborhood of $1$. Using the Cauchy property, you should be able to show that for some $j$, for all $i>j$, $u_i$ lies in a translation of $V$, hence in a compact space. so you can extract a converging subset from $u_i$. and when you can extract a converging subnet from a Cauchy net it has to converge to the limit of the extraction. (direct consequence of the Cauchy property)