SCHWEITZER 1998 Problem 10

Let $\xi_1 , \xi_2 , ...$ be a series of independent, zero-expected-value random variables for which $\lim_{n\to\infty} E(\xi_n ^ 2) = 0$ , and $S_n = \sum_{j = 1}^n \xi_j$ . Denote by I(A) the indicator function of event A. Prove that $$\frac{1}{\log n} \sum_{k = 1}^n \frac1k I\bigg(\max_{1\leq j\leq k} |S_j|>\sqrt k\bigg) \to 0$$ with probability 1 if $n\to\infty$ .