Let $ G $ be a metrizable group. If (i) $ K $ is a closed normal subgroup of $ G $ and (ii) both $ K $ and $ G/K $ are complete, then $ G $ is complete.
Here is how I am proceeding:
It can be assumed w.l.o.g. that the topology of $ G $ is induced by a right-invariant metric $ d $. Let $ (x_{n})_{n \in \mathbb{N}} $ be a right Cauchy sequence in $ G $. It suffices to show that some neighborhood $ V $ of $ e $ in $ G $ is complete. I don’t know how to proceed after this. Please help me.