measure theory - Intersections of measurable sets

Let $(X, \mathcal{A},\mu)$ be a measure space, and let $A_1,A_2,...\in \mathcal{A}$. Assume that $\sum_{j=1}^\infty \mu(A_j)<\infty.$ I have already proved following statements:

(1) $n \cdot \mu (\bigcap_{j=1}^\infty A_j)\leq \sum_{j=1}^\infty \mu(A_j) $

(2) $\lim_{n\to \infty} \mu (\bigcap_{j=1}^\infty A_j)=0$

(3) $ \mu (\bigcap_{j=1}^\infty A_j)=0$

I need to prove that if we set

$ E=\bigcap_{j=1}^\infty (\bigcup_{k=n}^\infty A_k)$

then i follows that $ \mu(E)=0$.

Can anyone help med with this problem.