Given $\mu$ is semifinite and $\mu(E)=\infty$, prove that if $C>0$, then $\exists$ measurable set $A \subseteq E$ such that $C < \mu(A) < \infty.$

Question

Let ($X,\Sigma, \mu)$ be a measure space. Suppose that $\mu$ is semifinite, $E \in \Sigma$, and $\mu(E)=\infty$. Prove that if $C>0$, then there exists some measurable set $A \subseteq E$ that satisfies $C < \mu(A) < \infty.$

I have a hint:

Define $C =$sup{ $\mu(F)$ $:$ $F \in \Sigma, F\subseteq E, \mu(F) < \infty$}. If $C<\infty$, then there exists measurable sets $F_k \subseteq E$ with finite measure such that $\mu(F_k)\to C$.

A quick definition: If every set $E \in \Sigma$ with $\mu(E)= \infty$ contains an $F \subseteq E$ such that $F \in \Sigma$ and $\mu(F) < \infty$, then $\mu$ is semifinite.

I know the first step is to find a contradiction if C is finite in the hint.