Let $E$ be a countable state space with $\sigma$-algebra $2^E$ and $X_t$ a Markov jump process with transition function
$$P_t(x,y) = \sum_{n=0}^\infty e^{-\lambda t}\frac{(\lambda t)^n}{n!}P^{(n)}(x,y),$$ where $P$ is a stochastic matrix. I'm trying to show that this is a Feller-Dynkin process, and am currently stuck on
$$\lim_{t \downarrow 0}P_tf(x) \rightarrow f(x).$$ As I understand it the left hand side should be \begin{align*} P_tf(x) &= \sum_{y \in E}f(y)P_t(x,y) \\ &= \sum_{y \in E}f(y)\sum_{n=0}^\infty e^{-\lambda t}\frac{(\lambda t)^n}{n!}P^{(n)}(x,y) \\ &= \sum_{n=0}^\infty e^{-\lambda t}\frac{(\lambda t)^n}{n!}\sum_{y \in E}f(y)P^{(n)}(x,y). \end{align*} This is where I'm stuck. We have that $e^{-\lambda t}\frac{(\lambda t)^n}{n!}$ sums to one over n, but each term is multiplied by different terms. Similarly $\sum_{y \in E}P^{(n)}(x,y)$ sums to one for constant $x,n$, but again we have each term multiplied by an $f(y)$ term.
Any ideas how to approach this?
Hint: Use
$$P_t f(x) = e^{-\lambda t} \underbrace{\sum_{y \in E} f(y) P^{(0)}(x,y)}_{f(x)} + t \underbrace{\sum_{n \geq 1} e^{-\lambda t} \frac{\lambda^n t^{n-1}}{n!} \sum_{y \in E} f(y) P^{(n)}(x,y)}_{\text{bounded (for e.g. $|t| \leq 1$)}}$$