I want to show that if $X \in L^1$, where $X$ is a real-valued random variable, the sum
$\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$
converges absolute.
My idea was the following:
Since $X \in L^1$, I have
$\sum_{n=1}^{\infty} n \cdot \mathbb{P}(|X|=n)~< \infty$.
So I get:
$\sum_{n=1}^{\infty}|\mathbb{P}(|X|>n)| = \sum_{n=1}^{\infty}\mathbb{P}(|X|>n) = \sum_{n=1}^{\infty} \sum_{k=n+1}^{\infty}\mathbb{P}(|X|=k) = \sum_{n=2}^{\infty}(n-1)\cdot \mathbb{P}(|X|=n) \leq \sum_{n=1}^{\infty} n \cdot \mathbb{P}(|X|=n)~<\infty$
This would only hold if $X$ could only attain natural numbers; how can I generalize this to show that
$\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$
converges absolute?
Thanks in advance.