Cauchy Completeness in elementary logic

The inverse of the distance between two structures ≢ ℬ of finite type τ is naturally measured by the smallest integer q such that a sentence of quantifier rank q - 1 is satisfied by but not by ℬ. In this way the space Strτ of structures of type τ is equipped with a pseudometric. The induced topology coincides with the elementary topology of Strτ. Using the rudiments of the theory of uniform spaces, in this elementary note we prove the convergence of every Cauchy net of structures, for any type τ.