Proof of "Dual normed vector space is complete" < As in the introduction of dual norm by Wiki, it says dual normed space $X'$ is always complete. How to prove that? or at least explain that? We all know the normed vector space is not always complete; if complete, all Cauchy sequences convergent. However, dual normed vector space is complete?

_Hint_ If $(f_n)$ is a Cauchy sequence of bounded functionals $f_n:V\to k$, take any $x$ and prove $f_n(x)$ is a Cauchy sequence in $k$. I'd take you assume your base field is complete, for example $k=\Bbb R$.