The following problem is a question from the book "Measure Theory and Probability" by Malcolm Adams and Victor Guillemin which I got Stuck in it:
Remember that a metric space it $\textit{complete}$ if every Cauchy sequence has a limit. Show that, with respect to the distance function $d(A,B)=\mu ^{*}(S(A,B))$, $2^X$ is complete.
P.S: by $\mu ^*(A)$ we mean the outer measure of $A$ , and by $S(A,B)$ we mean the symmetric difference $S(A,B)=(A-B) \cup (B-A)$.
I guess that every Cauchy sequence like $\{C_n\}$ will converge to the limit point $$c=\bigcap_{n=0}^{\infty} \bigcup_{n=0}^{\infty} C_n$$ However, I cannot prove it. Any help would be appreciated.