Hello,
I am trying to show that every metrizable locally compact topological group admits a complete metric generating the topology of the group
Hello, I am trying to show that every metrizable locally compact topological group admits a complete metric generating the topology of the group 


Every second countable, locally compact group admits a metric which is leftinvariant, generates the topology, and is proper (i.e. closed balls are compact). See Theorem 4.5 in http://arxiv.org/pdf/math/0606794 Such a metric is clearly complete. 

