CLT is true with conditions iff Lindeberg condition holds?

Question

In last lecture, we discussed about Lindeberg-Feller CLT. It was mentioned that CLT is true with two other conditions iff Lindeberg's condition holds. I can't verify this statement and I think the first condition is unnecessary. Can anyone help me?

Here are the two conditions:

1. $\lim_{n \to \infty}B_n=\infty$
2. $\lim_{n \to \infty}\frac{\max b_k^2}{B_n^2}=0$

where

$b_k^2=\operatorname{Var}(X_k)$

$B_n^2=\sum_1^n b_k^2$

and for reference: https://en.m.wikipedia.org/wiki/Lindeberg%27s_condition

Here is my note: There exists a positive number $\varepsilon$ s.t. $\max b_k^2>\varepsilon$. Thus applying condition2 we obtain condition1.