How to formalize this last step in my proof? Let $X_k \sim N(\xi, \sigma^2)$. Let $\xi > 0$. 
Consider $\frac{1}{k}X_k$, and the corresponding partial sums. I wish to show divergence to infinity, almost surely, of the partial sums.
I manage to show that $\sum_{k=1}^n X_k/k - \sum_{k=1}^n \xi/k$ converges to some finite value almost surely.
However, this can "obviously" only be true if the first term goes to infinity almost surely. If it is finite, or it oscillates, or whatever, the entire expression must converge to negative infinity, hence a contradiction. But what is the formal way to prove this last step?

The event $A=\{\sum_{k=1}^n X_k/k \to +\infty\}$ contains the event B = $\{\exists a, \sum_{k=1}^n X_k/k - \sum_{k=1}^n \xi/k \to a\}$. You've shown $P(B)=1$, therefore $P(A)=1$.
To show $B\subseteq A$, let $\omega\in B$, so
$$\sum_{k=1}^n X_k(\omega)/k - \sum_{k=1}^n \xi/k \to a$$
Now just work deterministically to show that $\omega\in A$. More generally, if $x_n$ and $y_n$ are sequences of real numbers, $y_n\to+\infty$ and $x_n-y_n\to a$, then we must have $x_n\to+\infty$, by applying additivity of limits to $x_n=(x_n-y_n)+y_n$.
