The problems arises from a consideration written on the book "Quantum computation and quantum information" from Michael A. Nielsen and Isaac L. Chuang on page 259.
In this chapter it's studied the Grover search algorithm as a quantum simulation and it's taken as a sample hamiltonian $H =|x\rangle\langle x| + |\psi\rangle \langle\psi|$ where $| x \rangle$ is the solution and $|\psi\rangle$ is the state we start from which is in the form: $$ |\psi\rangle=\sum_{x}\frac{|x\rangle}{\sqrt{N}} $$
The time evolution operator $e^{-i(|x\rangle\langle x| +|\psi\rangle\langle \psi|)t}$ is approximated for small $\Delta t$ as $$ U(\Delta t) = e^{-i|\psi\rangle\langle\psi|\Delta t}e^{-i| x\rangle\langle x|\Delta t} $$
and choosing $|y\rangle$ such that $\{|x\rangle, |y\rangle\}$ is an orthonormal base we take
$$ |\psi\rangle\doteq \begin{pmatrix} \alpha \\ \beta \end{pmatrix},|x\rangle\doteq\begin{pmatrix} 1 \\ 0 \end{pmatrix} $$ Then we represent the projectors $|x\rangle\langle{x}|$, $|\psi\rangle\langle\psi|$ in this base as $$ |\psi\rangle\langle\psi|=\frac{\mathbb{I}+\vec{\psi}\cdot\vec{\sigma}}{2} \hspace{1cm}\text{where}\hspace{1cm}\vec{\psi}= \begin{pmatrix} 2\alpha\beta \\ 0 \\ \alpha^2-\beta^2 \end{pmatrix} $$
$$ |x\rangle\langle x|=\frac{\mathbb{I}+\hat{z}\cdot\vec{\sigma}}{2} \hspace{1cm}\text{where}\hspace{1cm}\hat{z}= \begin{pmatrix} 0\\ 0 \\ 1 \end{pmatrix} $$
Calculating $U(\Delta t)$ in this base we obtain $$ U(\Delta t) = \left( \cos^2\left(\frac{\Delta t}{2}\right) - \sin^2\left(\frac{\Delta t}{2}\right)\vec{\psi}\cdot \hat{z} \right)\mathbb{I} - 2i\sin\left(\frac{\Delta t}{2}\right)\left(\cos\left(\frac{\Delta t}{2}\right)\frac{\vec{\psi}+\hat{z}}{2} + \sin\left(\frac{\Delta t}{2}\right)\frac{\vec{\psi}\, \times \, \hat{z}}{2} \right)\cdot\vec{\sigma} $$ And this relation should be valid for small $\Delta t$. Then the book says that a smart choice for a step would be $\Delta t = \pi$, because this would lead to the Grover iteration.
BUT doesn't choosing $\Delta t=\pi$ violate the approximation assumed in the beginning?
PS:
I tried to see if $$ U(\Delta t_1)U(\Delta t_2) = U(\Delta t_1 + \Delta t_2) $$ on a computer but it does not.