If $\mu$ is a complex measure and $\|\mu\| = \mu(X)$, then $\mu$ is a positive measure

Taken from Conway's A course in functional analysis Chapter 3 Section 7 Problem 2

Problem Statement: Let $X$ be a set and $\Omega$ a $\sigma$-algebra of subsets of $X$. Suppose $\mu$ is a complex-valued countably additive measure defined on $\Omega$ such that $\|\mu\| = \mu(X) < \infty$. Show that $\mu(\Delta) \geq 0$ for every $\Delta$ in $\Omega$

Given the section this problem is in, I assume we are supposed to use Banach limits to solve this problem. However, Banach limits are linear functionals acting on $l^\infty$ spaces which makes it hard for me to see how to apply it to this problem.