Krdytkyu

I'm stuck with the following problem:

Let $(X_k)_{k=1}^infty$ be a sequence of real-valued random variables. Show that there is a real sequence $(c_k)_{k=1}^infty$ such that
$$Pleft(lim_{krightarrowinfty}frac{X_k}{c_k}=0right)=1.$$

With the typical approach for proving almost sure convergence through the Borel-Cantelli-lemma you arrive at
$$Pleft(lim_{krightarrowinfty}frac{X_k}{c_k}neq 0right)leq sum_{n=1}^infty Pleft(limsup_{krightarrowinfty}|X_k|>frac{c_k}{n}right)$$
and it would be sufficient to find $(c_k)_{k=1}^infty$ such that
$$sum_{k=1}^infty Pleft(|X_k|>frac{c_k}{n}right)<infty.$$
If the $X_k$ were identically distributed, I think a construction of the form $c_k=kcdot q_{1-1/(k^2)}$ could work, where $q_alpha$ is an $alpha$-quantile of $|X_1|$, but I can't do this in the general case.

How can I solve the general case (no further assumption about the $X_k$)? Is Borel-Cantelli even a good approach here?

If $P{|X_k| >frac {c_k} {2^{k}}} <frac 1 {2^{k}}$ then $sum_k P{|X_k| >frac 1 {2^{k}}c_k} <infty$ and Borel Cantelli Lemma tells you that $frac {X_k} {c_k} to 0$ almost surely. You can always find $c_k$'s satisfying this condition, right? [All you need is the fact that $P{|X_k| >t} to 0$ as $t to infty$].