I'm studying Infinitely Divisible random variables using this [Lecture Notes][1]. And I have a question that is driving me crazy. 

In the proof of the "only if" part of the **Levy-Khintchine representation** (Theorem 5.3), the **Theorem 5.13** contains an essential step. In this Theorem 5.13, the truncation function $h'$ must satisfy conditions (\ref{hhdjj}):       
\begin{equation}\label{hhdjj}\tag{5.4}
h'(x)=1+o(|x|) \quad as\quad  |x|\to 0\quad  \hbox{ and }\quad h'(x)=O(1/|x|), \quad as \quad x\to\infty.
\end{equation}

However, in the proof of the "only if" part of the Theorem 5.3 (**page 17**), They approximate $\rho$ by a sequence of Compound Poisson random variables $\rho_n$ whose triple Levy-Khintchine representation and the use of Theorem 5.13 are described in the following print (Bottom of page 17):

[![enter image description here][3]][3]

So my big question is that they can't use Theorem 5.13 directly because $h\equiv 0$ doesn't satisfy the first condition in (\ref{hhdjj}). 

Am I making a faux pas by reading the notes?

I appreciate your help.


  [1]: https://actuarweb.aegean.gr/levy2019/uploads/1/1/5/5/115582233/baurdoux_papapantoleon_levyprocesses.pdf
  [3]: https://i.sstatic.net/FhN1F.png