Proof Completeness of a Metric Space with Complex Measure

Question

So I have a lot of trouble with this problem:

Let $M(X)$ be the set of signed measures on $(X, \mathcal A)$. We define a metric by $d(\mu_1,\mu_2)=|\mu_1-\mu_2|(X)$. Proof that the metric space $(M(X),d)$ is complete.

So my original approach was to create a Cauchy Sequence $\{\mu_n\}_{n \in \mathcal N}$, but my issue is that I have trouble minimizing $d(\mu_n,\lim_{k\to\infty}(\mu_k))$ (I have no idea how to formalize $\lim_{k\to\infty}(\mu_k)$). This is where I am stuck at, any hint is welcomed!

Answer 1

Hints:

(1) Show $(\mu_n(A))_n$ is Cauchy for a measurable set $A$.

(2) Define $\mu:=\lim_n \mu_n$ (pointwise limit). By (1), this is well-defined.

(3) Show $\mu_n\to \mu$.