How can show that that $\lim_\limits{k \to \infty} \sup\limits_{i \in J} E[(|Y_i|-k)^+]=0$ implies uniform integrability of the set of r.v's $(Y_i)_{i \in J}$
I have spent quite some time ,unsuccessfully trying to show that the above statement implies
$$\lim_{k \to \infty} \sup_{i \in J} E[|Y_i| \mathbb{1}_{\{|Y_i|\geq k\}}]=0.$$
It is clear to me that
$E[|Y_i| \mathbb{1}_{\{|Y_i|\geq k\}}] \geq E[(|Y_i|-k)^+]$ so applying sup and lim to both sides of the inequality changes nothing but since
$$\lim_{k \to \infty} \sup_{i \in J} E[(|Y_i|-k)^+]=0,$$
$\lim\limits_{k \to \infty} \sup\limits_{i \in J}E[|Y_i| \mathbb{1}_{\{|Y_i|\geq k\}}] \geq 0$ is not necessarily $0$ I tried several other things but I was unable to prove it. Can someone help me?