Proving $n^{-1/2}\sum_{k=1}^{n}a_{nk}X_k\to0$ a.s

I encountered an exercise while learning the law of large numbers.

$\{X_n;n\ge 1\}$ is a sequence of independent and identically distributed random variables.Prove that $EX_k=1,EX_{1}^{2}<\infty \iff$ For any sequence $\{a_{nk};k=1,\cdots,n,n\ge1\}$ that satisfies the condition $\sum_{k=1}^{\infty}a_{nk}^{2}\to1$, $n^{-1/2}\sum_{k=1}^{n}a_{nk}X_k\to0$ a.s is established.

The conditions are so simple that I don’t know how to prove it using only the existence of the second moment. Welcome to give ideas on how to solve it.

The second moment of $\sum_{k=1}^n a_{nk} X_k$ equals $\sum_{k=1}^n a_{nk}^2,$ so, when rescaled by $n^{-1/2}$ does converge to zero, but that is not enough. If the first condition is as you suggest, then the mean of $n^{-1/2} \sum_{k=1}^n a_{nk}X:k$ does not go to zero. For example, you can set $a_{nk} = n^{(-1/2)}.$ Since neither the mean nor the variance goes to zero, the convergence does not hold. It is possible that the first condition is something different...