If $N$ is a normed a linear space, then its dual vector space $N^\*$ is always complete.

Attempt: Let $\{f\_n\}$ be a Cauchy sequence in $N^\*$. Then, for some *$\varepsilon > 0$*, there exists $m,n \in \mathbb{N}$ such that $\|f\_n - f\_m \| < \varepsilon$. The way to show that the limit lies in $N^\*$ would be to show that $f\_n(x)$ converges in $K = \mathbb{C}$ or $\mathbb{R}$. How do I show that the limit of linear functionals is still a linear functional?