I'm studying Infinitely Divisible random variables using this Lecture Notes. 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):

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.