How to prove that if $\mu$ is finite, then it is sum-finite Im having some trouble with how to prove that if $\mu$ is finite then it is sum-finite.

I know that if $\mu$ is finite on a measurable space (E,A) if $\mu<\infty$ and that $\mu$ is sum-finite on a measurable space (E,A) if $\mu=\sum_{n=1}^{\infty} \mu_n$, where $\mu_n$, for $n\in \mathbb{N}$, is finite measures.
But I dont know how to tie these definitions together.

Hint: The trivial measure $\zeta$ that assigns 0 measure to all sets is finite.
Solution:

 $$ \mu = \mu + \sum_{n=2}^\infty\zeta $$ is an expression for $\mu$ as a countable sum of finite measures.

