Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of real i.i.d. random variables with $\mathbb{E}(X_1)=\infty$ and $(a_n)_{n\in\mathbb{N}}$ a positive real sequence such that $\frac{a_n}{n}$ is non-decreasing in $n$. I want to prove that the following is equivalent: $$ \sum_{n=1}^{\infty}P(|X_n|>a_n)<\infty \iff P(\limsup\limits_{n\to\infty}|S_n|/a_n<\infty)=1 $$
My attempt:
Apparently this is a direct consequence of a paper of a Theorem from Fellers paper "A Limit Theorem for Random Variables with Infinite Moments" from 1946.
The Theorem in the paper says:
Let $(X_n)_{n\in\mathbb{N}}$ and $(a_n)_{n\in\mathbb{N}}$ be given as above, then $$ P(|S_n|>a_n \text{ i.o.})=0 \text{ if }\sum_{n=1}^{\infty}P(|X_n|>a_n)<\infty $$ and $$ P(|S_n|>a_n \text{ i.o.})=1 \text{ if }\sum_{n=1}^{\infty}P(|X_n|>a_n)=\infty $$
I can prove that $\sum_{n=1}^{\infty}P(|X_n|>a_n)<\infty \Rightarrow P(\limsup\limits_{n\to\infty}|S_n|/a_n<\infty)=1$, but I'm lost for the other direction. One of my ideas was the following:
Let $c\geq1$. If $\sum_{n=1}^{\infty}P(|X_n|>a_n c)=\infty$ then according to the Theorem cited above we have $$ 1=P(|S_n|>a_nc \text{ i.o.})=P\Bigl(\frac{|S_n|}{a_n}>c \text{ i.o.}\Bigr). $$ So if we let $c\to \infty$, we can find a random subsequence $(n_k)_{k\in\mathbb{N}}$ such that $\lim\limits_{k\to\infty}|S_{n_k}|/a_{n_k} = \infty$ a.s. and hence $P(\limsup\limits_{n\to\infty}|S_n|/a_n=\infty)=1$. Using Kolmogorov's zero-one law we get that $P(\limsup\limits_{n\to\infty}|S_n|/a_n<\infty)<1$ implies $P(\limsup\limits_{n\to\infty}|S_n|/a_n<\infty)=0$ and hence we arrive at a contradiction.
However the problem is that I can't say for sure that $\sum_{n=1}^{\infty}P(|X_n|>a_n c)=\infty$, since for $c\geq 1$ it only holds $\sum_{n=1}^{\infty}P(|X_n|>a_n c)\leq \sum_{n=1}^{\infty}P(|X_n|>a_n)=\infty$.
Any advice would be much appreciated!