the projections involved are finite-dimensional.

\[fin\] In addition to the hypotheses of Theorem \[main\], assume that the orthogonal projection $P$ as well as $P(t)$ is of finite dimension. Then\
\[.2em\] (i) the formulae (\[product-s\])–(\[product-ns2\]) hold in the norm of ${\cal H}$ as $n\rightarrow \infty$, uniformly on each compact interval of the variable $t$ in ${\mathbb{R}}\setminus \{0\}$,\
\[.1em\] (ii) it also holds for $\varepsilon = \pm 1$ that as $n\rightarrow
\infty$, $$\begin{aligned}
&& [P(t/n)\exp(-i\varepsilon tH/n) P(t/n)]^{n}
 \longrightarrow
 \exp(-i\varepsilon tH_P)\,P\,, \label{} \\
&& [P(t/n)\,\exp(-i\varepsilon tH/n)]^{n}
 \longrightarrow
\exp(-i\varepsilon tH_P)\,P\,, \label{}\\
&& [\,\exp(-i\varepsilon tH/n)\,P(t/n)]^{n}
 \longrightarrow
\exp(-i\varepsilon tH_P)\,P\,, \label{}
 \end{aligned}$$ strongly on ${\cal H}$, uniformly on each compact interval in the variable $t\in{\mathbb{R}}$.

Before proving Theorems \[main\] and \[fin\] and Corollary \[cor2\] let us comment briefly on several other aspects of the result.

\[phys-conv\] [While the necessity to pick a subsequence makes the pointwise convergence result weaker than desired, let us notice that from the physical point of view the convergence in $L_\mathrm{loc}^2({\mathbb{R}}; {\cal H})$ can be regarded as satisfactory. The point is that any actual measurement, in particular that of time, is burdened with errors. Suppose thus we perform the Zeno experiment on numerous copies of the system. The time value in the results 