Show that $\limsup_n \frac{S_n}{n(\log \log n)^{1-\epsilon}} = \infty$ with probability 1

Question

Let $\{X_k\}$ be independent random variables such that $\mathbb{P}(X_k = k) = 1/k$ and $\mathbb{P}(X_k = 0) = 1-1/k$. Consider the sum $S_n = \sum_{k=1}^{n} X_k$. Show that for any $\epsilon\in(0,1)$, with probability $1$, $$ \limsup_n \frac{S_n}{n(\log \log n)^{1-\epsilon}} = \infty. $$ I am trying to use Borel-Cantelli, but I have trouble constructing a sequence of independent events. Any help will be appreciated.