To prove: Let $S$ be an arbitrary, non-empty set and let $\Sigma_0$ be an algebra on $S$. Let $\mu:\Sigma_0 \rightarrow [0,\infty]$ be a countably additive map. Show that for every decreasing sequence of sets $H_n \in \Sigma_0$ with $\cap_{n=1}^\infty H_n = \varnothing$ we have that $\lim_{n \rightarrow \infty}\mu(H_n) = 0$.
My proof:Let $H_n$ to be a decreasing sequence of sets in $\Sigma_0$ with $\bigcap_{n=1}^\infty H_n = \varnothing$. Define $H'_n:= H_n \setminus H_{n+1}$. These $H'_n$ can be seen as the disjunct components of our sequence. Then we have $H_n = \bigcup_{k=n}^\infty H'_k$. Therefore some manipulation gives: \begin{align*} \mu(H_n) &= \mu\left( \bigcup_{k=n}^\infty H'_k \right) \\ &= \sum_{k=n}^\infty \mu(H'_k) \\ &= \sum_{k=n}^\infty \mu(H_k \setminus H_{k+1}) \\ &= \sum_{k=n}^\infty \mu(H_k) - \sum_{k=n}^\infty \mu(H_{k+1}). \end{align*}
If we then let $n \rightarrow \infty$ we see that the right hand side goes to zero, which shows that $\lim_{n \rightarrow \infty} \mu(H_n) = 0$.
My problem is I haven't used that $\cap_{n=1}^\infty H_n = \varnothing$, which is pretty essential. What am I doing wrong?!