Completeness of locally $k_\omega$-groups and related infinite-dimensional Lie groups

Recall that a topological space is said to be a $k_\omega$-space if it is the direct limit of an ascending sequence of compact Hausdorff topological spaces. If each point in a Hausdorff space $X$ has an open neighbourhood which is a $k_\omega$-space, then $X$ is called locally $k_\omega$. We show that a topological group is complete whenever the underlying topological space is locally $k_\omega$. As a consequence, every infinite-dimensional Lie group modelled on a Silva space is complete.