This seems to me that it should be simple, but somehow it elludes me:
Let $(X,\mathcal{F},\mu)$ be a measure space such that for all $\epsilon>0$, there exists $A\in \mathcal{F}$ satisfying $0<\mu(A)< \epsilon$. Construct a disjoint sequence of measurable sets $A_n$ such that $\mu(A_n)\rightarrow0$ and $\mu(A_n)>0$ for all $n\in \mathbb{N}$.
Now it is relatively simple to construct such a sequence without demanding that $\mu(A_n)>0$. But I am having trouble giving a recursive construction which ensures that the measures are strictly positive. It feels to me that it should be a simple argument which I am missing.
First obtain a set $E_1$ of finite positive measure say of measure $\delta_1$. Now, use the given condition to get another set $E_2$ of measure $0<\delta_2<\delta_1/2$. Keep on doing this to get a sequence of sets $\{E_i\}$ of measures $\delta_i$ such that $0<\delta_{i+1}<\delta_i/2$. Now, let $F_i=E_{i}-\bigcup_{i+1}^{\infty} E_{j}$. These are disjoint sets of positive measure whose measure tends to $0$. They are of positive measure because $1=\sum_1^\infty 1/2^i$.