Metric space and dense subset
Posted: Wed Mar 09, 2016 6:12 pm
Let \(\displaystyle{\left(X,d\right)}\) be a metric space and \(\displaystyle{D}\) a dense subset of \(\displaystyle{X}\) having the
property : Each \(\displaystyle{\rm{Cauchy}}\) sequence of elements of \(\displaystyle{D}\) converges to \(\displaystyle{X}\) .
Prove that the metric space \(\displaystyle{\left(X,d\right)}\) is complete.
Comment : The metric space \(\displaystyle{\left(\mathbb{R},|\cdot|\right)}\) with \(\displaystyle{D=\mathbb{Q}}\) is an example of such space.
property : Each \(\displaystyle{\rm{Cauchy}}\) sequence of elements of \(\displaystyle{D}\) converges to \(\displaystyle{X}\) .
Prove that the metric space \(\displaystyle{\left(X,d\right)}\) is complete.
Comment : The metric space \(\displaystyle{\left(\mathbb{R},|\cdot|\right)}\) with \(\displaystyle{D=\mathbb{Q}}\) is an example of such space.