  We define a metric ultraproduct of topological groups with left-invariant
metric, and show that there is a countable sequence of finite groups with
left-invariant metric whose metric ultraproduct contains isometrically as a
subgroup every separable topological group with left-invariant metric.
  In particular, there is a countable sequence of finite groups with
left-invariant metric such that every finite subset of an arbitrary topological
group with left-invariant metric may be approximated by all but finitely many
of them.
  We compare our results with related concepts such as sofic groups,
hyperlinear groups and weakly sofic groups.
