Does uniformly bounded second moment imply Lindeberg's condition>

Question

Suppose an independent sequence $X_i \in \mathbb{R}^k$ is such that $\sup_n E(\Vert X_i \Vert^2) \leq M$ for some constant $M$. Does this imply that $X_n$ satisfies the following Lindeberg-type condition: \begin{align*} \frac{1}{n}\sum_{i=1}^nE\left( \Vert X_i \Vert^2 I\{\Vert X_n \Vert \geq \epsilon \sqrt{n}\} \right) \rightarrow 0 \end{align*}

It seems clear to me that the sum does not diverge, since it is bounded by $M$, but is it true that it goes to zero?

Answer 1

The answer is No.
Try $X_n= n.\mathcal{Bernoulli}(1/n^2)$