# Why are locally compact groups Weil complete?

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.

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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)
Exactly as you would do with a sequence on a metric space : You have $(u_i)$ a cauchy net and $(u_j)$ a subnet converging to $u$. Take $V$ a neighborhood of $u$, and $V'$ a neighborhood of $0$ such that if $(u-x)$ and $(x-a)$ are in $V'$ then $a$ is in $V$. write the cauchy propertie for $V'$, then write that there is therm of $(u_j)$ which is $V'$ close to $u$ for $j$ big enough to be converne bu the previous cauchy propertie you wrote, you can deduce from those two thing, that for any $i$ concerne bu the cauchy properties, $u_i$ is in $V$. – Simon Henry Oct 6 '12 at 8:26