Uniform metric
= Uniform metric
{title2=$d_\infty(f,g)=\sup_xd(f(x),g(x))$}
The supremum of pointwise distances defines a metric on maps when it is finite, for example if the target <metric space> is bounded. Convergence in this metric is <uniform convergence>. If the target is complete, Cauchy sequences have pointwise limits and the Cauchy estimate makes convergence uniform.