Claim 1: $q\left ( i,j \right )=\lambda u\left ( i,j \right )$
where $q\left ( i,j \right )$ is the jump rate for a Markov chain at initial state i to a final state j in one step and $u\left ( i,j \right )$ is the associated transition probability for a Markov chain at an initial state i to a final state j in one step.
Claim 2: For a Markov chain at initial state n and a final state n+1, the jump rate is $\lambda$, for all n $\geq 0$
Claim 2 implies that the transition probability from a state i to j in one step is one. How can I verify this?
Excerpt from Essential of Stochastic Processes by Durrett.
