I need help as soon as possible to understand the following :

For any non-negative random variable ζ : $\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)\leq\mathbb{E}(\zeta)\leq 1+\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)$.

Therefore, if $x_0,\dots,x_n$ are iid random variables and $\mathbb{E}(\ln{(1+|x_0|))}<\infty$, it follows that : $\sum_{k=1}^{\infty}\mathbb{P}(|x_k|\geq e^{\gamma k})<\infty$ for any positive constant $\gamma$.

I don't understand where the $\gamma>0$ is coming from? Isn't if for $\gamma\geq 1$?

Thanks !!

I need help to understand the following :

For any non-negative random variable $\zeta$: $\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)\leq\mathbb{E}(\zeta)\leq 1+\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)$.

Therefore, if $x_0,\dotsc,x_n$ are iid random variables and $\mathbb{E}(\ln{(1+|x_0|))}<\infty$, it follows that : $\sum_{k=1}^{\infty}\mathbb{P}(|x_k|\geq e^{\gamma k})<\infty$ for any positive constant $\gamma$.

I don't understand where the $\gamma>0$ is coming from? Isn't it for $\gamma\geq 1$?