The question doesn't make sense, or alternatively it is true by
definition.
A Hilbert Space is a complete inner product space - complete in
the metric induced by the norm defined by the inner product over
the space.
In other words an inner product space is a vector space with an
inner product defined on it.
An inner product then defines a norm on the space, and every
norm on a space induces a metric.
A Hilbert Space is thus also a complete metric space, simply
where the metric is induced by the inner product.