A equivalent definition of the Feller Process. I saw this on Liggett's Book (P.95).
Let $S=%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,$ and suppose $\left( X_{t}\right) _{t\geq 0}$ is a continuous-time Markov
process with state space $S$ and transition function $\left( p_{t}\right)
_{t\geq 0}.$ Show that $\left( X_{t}\right) _{t\geq 0}$ is a Feller process
if and only if
$$
\lim_{t\downarrow 0}p\left( x,\left\{ x\right\} \right) =1\text{ for all }%
x\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$$
and
$$
\lim_{x\rightarrow \infty }p_{t}\left( x,\left\{ y\right\} \right) =0\text{
for all }y\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\text{ and }t>0.
$$
 A: Only if:
Suppose $X$ is Feller. Take $x\in\mathbb{N}$. For $y\in\mathbb{N}$, let $f(y) = \delta_{x,y}$.
$$\begin{align}\lim_{t\to 0}p_t(x, \{x\}) &= \lim_{t\to 0}\mathbb{E}_x[f(X_t)]\\
&= f(x)\\
&= 1\end{align}$$
Fix $y\in\mathbb{N}$ and $t>0$. Suppose 
$$\lim_{x\to\infty}p_t(x,\{y\}) > 0.$$
Then there is $\epsilon>0$ and an infinite subset $A$ of $\mathbb{N}$ such that, for $a\in A$, $p(a, \{y\})\ge \epsilon$. For $x\in\mathbb{N}$, let $g(x) =\delta_{x,y}$. $g\in C(\mathbb{N})$.
$$\begin{align}\mathbb{E}_x[g(X_t)] &= p_t(x, {y})\\
&\ge \epsilon,\end{align}$$
So $p_tg\notin C(\mathbb{N})$, contradicting the Feller property of $X$.
If: Suppose $X$ is a process for which the given pair of properties holds. Take $f\in C(\mathbb{N})$. Fix $x\in\mathbb{N}$. 
$$\begin{align}\lim_{t\to 0}E_x[f(X_t)] &= \lim_{t\to 0}\mathbb{E}_x\left[\sum_{y\in\mathbb{N}}f(y)\delta_{X_t,y}\right]\\
&= \lim_{t\to 0} \sum_{y\in\mathbb{N}}\mathbb{E}_x[f(y)\delta_{X_t,y}]\\
&=\lim_{t\to 0} \sum_{y\in\mathbb{N}}f(y)p_t(x, y)\\
&= \sum_{y\in\mathbb{N}}f(y)\delta_{x,y}\\
&= f(x),\end{align}$$
with the interchanges being justified by boundedness of $f$.
Let $M<\infty$ bound |f|. Fix $\epsilon > 0$ and let $L$ be the compact set on which $|f|> \epsilon / 2$. Fix $x\in\mathbb{N}$.
$$\lim_{x\to \infty}p(x, L) = 0.$$
$$\begin{align}\mathbb{E}_x[f(X_t)] &\le Mp(x, L) + \epsilon(1 - p(x, L)) / 2\\
&\le Mp(x, L) + \epsilon / 2\end{align}$$
We may choose $N$ such that $Mp(x,L) \le \epsilon / 2$, for $x\ge N$. Then, outside of the compact set $\{0, \cdots, N\}$, $x\mapsto\mathbb{E}_x[f(X_t)]\le \epsilon$. So $X$ is Feller. 
