A Poisson process with intensity $\lambda > 0$ is a random function $t \mapsto N(t)$ with domain $[0,\infty)$ and taking values in $\mathbb{Z}_{\geq0}$ such that
i) N(0) = 0;
ii) if $s\leq t$ then $N(s)\leq N(t)$;
iii) for every $t\geq0$, as $h\downarrow0$,we have $$P(N(t +h) = j|N(t) = i) =\begin{cases} 0 &\text{if } j < i \\ 1−λh+o(h) &\text{if } j=i \\ λh+o(h) &\text{if } j = i+1 \\ o(h) &\text{if } j>i+1 \end{cases};$$
iv) If $s < t$ then $N(s)$ and $N (t ) - N (s)$ are independent.
I have seen this definition of a Poisson process but am not fully convinced that it is really equivalent to the usual ones. In particular, I don't see how it follows from this that for example $$\lim_{h\downarrow 0} P(N(t) = i|N(t-h) = i) = 1,$$ which can be easily proven with the definition that includes Poisson increments. Does also this equality follow somehow from iii)?