I am going through a chapter on Poisson processes in Achim Klenke's Probability Theory textbook. He defines the following conditions that a process which counts the number of clicks of a Geiger counter in a fixed time interval should have. 
To explain condition (P5), he defines $\lambda = \limsup_{\varepsilon \downarrow 0} \varepsilon^{-1} \mathbf{P}[N_\varepsilon \geq 2]$. Then he says that for any $n \in \mathbb{N}$ and $\varepsilon > 0$, we have
$$ \label{eq:1} \mathbf{P}[N_{2^{-n}} \geq 2] \geq \lfloor {2^{-n}/\varepsilon} \rfloor \mathbf{P}[N_\varepsilon \geq 2] - {\lfloor 2^{-n}/\varepsilon \rfloor}^2 {\mathbf{P}[N_\varepsilon \geq 2]}^2, $$ and then concludes $$ 2^n \mathbf{P}[N_{2^{-n}} \geq 2] \geq \lambda - 2^{-n}\lambda^2 \to \lambda \hspace{0.5cm} \text{as} \hspace{0.5cm} n \to \infty. $$ The way I justify the first item is by noting $$ \mathbf{P}[N_{2^{-n}} \geq 2] = \mathbf{P}[N_{(2^{-n}/\varepsilon)\varepsilon} \geq 2] \geq \mathbf{P}[N_{\lfloor 2^{-n}/\varepsilon \rfloor \varepsilon} \geq 2] = \lfloor {2^{-n}/\varepsilon} \rfloor \mathbf{P}[N_\varepsilon \geq 2], $$ which would then be obviously greater equal the RHS of the first inequality above. The second equality follows from (P1) and (P2). So I have two questions
(i) Is my reasoning for the first inequality correct? If not, what is my mistake and what is the correct reasoning?
(ii) If yes, why is the "square" term needed? I suspect it is for the second inequality, but I am not sure why.
Klenke then goes on to explain the condition with further steps, but those I understand, conditional on the above. Thanks in advance!
The equality you wrote: $$ \mathbf{P}[N_{\lfloor 2^{-n}/\varepsilon \rfloor \varepsilon} \geq 2] = \lfloor {2^{-n}/\varepsilon} \rfloor \mathbf{P}[N_\varepsilon \geq 2], $$ is not correct. The LHS contains the union of $\lfloor 2^{-n}/\varepsilon \rfloor$ independent events of probability $\mathbf{P}[N_\varepsilon \geq 2]$ each, and Klenke uses the first Bonferroni inequality (discussed in [1], with $k=2$) to give a lower bound for the probability of the union.
[1] https://en.wikipedia.org/wiki/Boole%27s_inequality