Prove that there exists a numeric sequence $c_1, c_2,...$ such that $\xi_n / c_n \to 0$ almost surely as $n \to \infty$. Let $\xi_1, \xi_2,...$ be a sequence of random variables defined on the same probability space. Prove that there exists a numeric sequence $c_1, c_2,...$ such that $\xi_n / c_n \to 0$ almost surely as $n \to \infty$.
My attempt: I guess the proof needs First Borel-Cantelli Lemma. But I don't know how to prove.

Every random variable $X$ is tight so that for every $\epsilon >0$ there exists a $m$ such that:
$$P(X\geq m) \leq \epsilon.$$
So for a given sequence $X_i$ we can choose a sequence $m_i$ such that
$$\sum_i^\infty P(X_i\geq m_i)<\infty.$$
So by the Borel-Cantelli lemma,
$$P(\sup_i X_i\geq m_i) =0.$$
Now choose a sequence $a_i=m_i \epsilon$ and apply the above result to this sequence.
