real analysis - ${f_n}$ is uniformly integrable if and only if $sup_n int |f_n|,dmu < infty$ and ${f_n}$ is uniformly absolutely continuous?

Let $(X, \mathcal{A}, \mu)$ be a measure space. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon$ there exists $M$ such that$$\int_{\{x : |f_n(x)| > M\}} |f_n(x)|\,d\mu < \epsilon$$for each $n$. The sequence is uniformly absolutely continuous if given $\epsilon$ there exists $\delta$ such that$$\left|\int_A f_n\,d\mu\right| < \epsilon$$for each $n$ if $\mu(A) < \delta$.

Suppose $\mu$ is a finite measure. How do I see that $\{f_n\}$ is uniformly integrable if and only if $\sup_n \int |f_n|\,d\mu < \infty$ and $\{f_n\}$ is uniformly absolutely continuous?