Show that if $\{f_n\}_{n\geq 1}$ is a uniformly integrable family of functions, then $\lim_{t\to\infty}\int|f_n|I_{|f_n|>t} = 0$ for all $n$.
I feel like this should fall right out of the definition of uniform integrability, but it's not coming to me for some reason.
The definition of uniform integrability I'm working with is;
$\{f_n\}_{n\geq1}$ is uniformly integrable if for all $\epsilon>0$ there is a $\delta>0$ such that if $\mu(A)<\delta$ then $\int_A|f_n|<\epsilon$, for all $f_n$.
It's not clear to me why I should be able to get $\mu\left(\{x:|f_n(x)|>t\} \right)<\delta$ for every $n$.
Any thoughts are appreciated.
Thanks in advance.