# Equivalent definition for uniform integrability.

I have seen these two definitions for uniform integrability, and I want to show that they are equivalent.

Definition 1

A collection of random variables $$X_i, i \in I$$ is uniformly integrable if for every $$\epsilon>0$$ there exists a $$K$$ such that $$E[|X_i|\cdot1_{|X_i|\ge K}]\le \epsilon,$$ for all $$i \in I$$.

Definition 2

A collection of random variables $$X_i, i \in I$$ is uniformly integrable, if there exists an $$M$$ such that

$$E[|X_i|]\le M,$$ for all $$i \in I.$$ And if for every $$\epsilon>0$$, there exists a $$\delta$$ so that if $$P(A)<\delta$$, then $$E[|X_i|\cdot 1_A]\le \epsilon,$$ for all $$i \in I.$$

Do you see how to prove this? I am able to show that definition 1 gives us a bound $$M$$ on the expectation, but the rest of the implication I am not able to show. I am also not able to go from definition 2 to definition 1.