Assume that $X_t$ is a continuous time Markov-chain with finite state space, and assume it is regular. That is:
If $t_1<t_2\ldots <t_n $ then
$P(X_{t_n}=i_n|X_{t_1}=i_1\cap X_{t_2}=i_2\cap\ldots \cap X_{t_{n-1}}=i_{n-1})=P(X_{t_n}=i_n| X_{t_{n-1}}=i_{n-1})$.
Define $P_{ij}(a,b)=P(X_b=j|X_a=i)$.
For $i\ne j$ the quantity
$\lim\limits_{\Delta t\rightarrow0^+}\frac{P_{ij}(t,t+\Delta t)}{\Delta t}=\mu_{ij}(t)$, exists and is continuous in $t$.
We also define
$\nu _i(t)=\lim\limits_{\Delta t\rightarrow0^+}\frac{1-P_{ii}(t,t+\Delta t)}{\Delta t}$.
Then some books say that the probability of two or more transitions in an intervel of length $\Delta t$ is $o(\Delta t)$. But how exactly is this proved?