Showing almost sure divergence Doing some exercises as preparation for an upcoming exam, but im sort of stuck at this exercise: \
Assume that $X_1,X_2,...$ is an i.i.d sequence, such that $X_1 \sim \mathcal{N} (\xi , \sigma^2)$, with $\xi>0$  . Define
$$
S_n=\sum_{k=1}^n \frac{X_k}{k}
$$
Show that $X_n \to \infty $ almost surely as n tends to infinity.
Im not aware of any theorem that can ease my way of showing this quickly. 
But im trying something going this direction:
$$
X_n \stackrel{a.s.}{\to} \infty  \iff \forall \varepsilon>0 : P(S_n>\varepsilon  \quad evt.)=1 \iff \forall \varepsilon>0: P( S_n \leq \varepsilon  \quad i.o.)=0
$$
$$
\Leftarrow \forall \varepsilon>0 : \sum_{k=1}^\infty P(S_n \leq \varepsilon) <\infty
$$
And from here i dont know where to go, because i have no closed form expression of the above probabilities, or even any idea if it holds.
Any tips/tricks/solutions are welcome.
 A: Hint: $X_k/k \sim \mathcal{N}(\xi/k, \sigma^2/k^2)$. Then, $S_n \sim \mathcal{N}(\xi\sum_{k=1}^n\frac{1}{k},\sigma^2 \sum_{k=1}^n \frac{1}{k^2})$. The mean grows to infinity, but the variance is at most $\sigma^2 \frac{\pi^2}{6}$.
A: Let's begin with a result that has nothing to do with probability.
If $(x_k)$ is a sequence of numbers where $\sum_{k=1}^n {x_k\over k}$ converges
to a finite limit $\alpha$, then ${1\over n}\sum_{k=1}^n x_k\to 0$. 
Proof: Note that   $\sum_{k=1}^n {x_k\over k}\to\alpha$ implies 
that the Cesàro averages also converge to $\alpha$: 
$${1\over n}\sum_{j=2}^n \left( \sum_{k=1}^{j-1}{x_k\over k}\right)\to\alpha. $$
But since 
$$\begin{eqnarray*}
{1\over n}\sum_{j=2}^n \left( \sum_{k=1}^{j-1}{x_k\over k}\right) 
 &=&{1\over n}\sum_{k=1}^n \left( \sum_{j=k+1}^n {x_k\over k}\right)\\[5pt] 
 &=&\sum_{k=1}^n{x_k\over k}-{1\over n}\sum_{k=1}^n x_k.
 \end{eqnarray*}$$
we deduce that  ${1\over n}\sum_{k=1}^n x_k\to 0$.
Now, let's put probability back into the picture. 
The law of large numbers gives ${1\over n}\sum_{k=1}^n X_k\to\xi>0$
 and therefore  $\sum_{k=1}^n {X_k\over k}$ cannot converge to a finite value. 
