Suppose that $\left\{X_{n,j} : n ∈ N, j = 1, \dots , k_n\right\}$ is a triangular array that satisfies the hypotheses of the Lindeberg-Feller central limit theorem. In particular, $E(X_{n,j}) = 0$ and $\sum_{j=1}^{k_n}E(X^2_{ n,j} ) = 1.$ Show that the array is infinitesimal, that is, $\lim_{n→∞} \max\{P(|X_{n,j} | > ε : j = 1, . . . , k_n\} = 0$ for every $ε > 0$. In particular, conclude that $\lim_{n\rightarrow \infty} k_n = +∞.$
My Try:
$$\{P(|X_{n,j} | > ε : j = 1, . . . , k_n\}=\Pi_{j=1}^{k_n} \{P(|X_{n,j} | > ε\}\leq \frac{1}{((\epsilon n)^2)^{k_n}} \Pi_{j=1}^{k_n}E(X^2_{ n,j} )$$ By the hypotheses I know that $E(X^2_{ n,j} )$'s are positive and strictly less than $1$. So, I understand why we must prove $\lim_{n\rightarrow \infty} k_n = +∞.$. But I am stuck how to prove it. Any help please.
Notice that for all $j\in\left\{1,\dots,k_n\right\}$, $$\mathbb P\left\{\max_{1\leqslant j\leqslant n}\left\lvert X_{n,j}\right\rvert \gt \varepsilon \right\}\leqslant \sum_{j=1}^n\mathbb P\left\{ \left\lvert X_{n,j}\right\rvert \gt \varepsilon \right\}\leqslant\frac 1{\varepsilon^2}\sum_{j=1}^n \mathbb E\left[X_{n,j}^2 \mathbb 1\left\{\left\lvert X_{n,j}\right\rvert \gt \varepsilon \right\}\right].$$