I am a bit confused on proving the Markov property for Poisson processes. That is, we want to prove, if $X = (X_t: t \in \mathbb{R})$ is a Poisson process with rate $\lambda$:
$P(X_{t_n} = a_n | X_{t_{n-1}} = a_{n-1}, X_{t_{n-2}} = a_{n-2},...) = P(X_{t_n} = a_n | X_{t_{n-1}} = a_{n-1})$ for $\quad a_1 \leq a_2 \leq ... \leq a_n$ and $\quad t_1 \leq t_2 \leq ... \leq t_n$
What I have is:
$N_n = X_{t_n} - X_{t_{n-1}}$~ Poisson($\lambda (t_n-t_{n-1})$)
$N_{n-1} = X_{t_n} - X_{t_{n-2}}$~ Poisson($\lambda (t_n-t_{n-2})$)
$N_{n-2} = X_{t_n} - X_{t_{n-3}}$~ Poisson($\lambda (t_n-t_{n-3})$)
.... with $N_i$ independent
Then, $P(X_{t_n} = a_n | X_{t_{n-1}} = a_{n-1}, X_{t_{n-2}} = a_{n-2} ,...) = P(N_n = a_n, N_{n-1} = a_{n-1}, N_{n-2} = a_{n-2},...) = \frac{e^{-\lambda (t_n-t_{n-1})}*(\lambda (t_n-t_{n-1}))^{a_n-a_{n-1}} }{{a_n-a_{n-1}}!}*...*\frac{e^{-\lambda (t_n-t_1)}*(\lambda (t_n-t_1))^{a_n-a_1} }{{a_n-a_1}!}$
While $P(X_{t_n} = a_n | X_{t_{n-1}} = a_{n-1}) = P(N_n = a_n-a_{n-1}) = \frac{e^{-\lambda (t_n-t_{n-1})}*(\lambda (t_n-t_{n-1}))^{a_n-a_{n-1}} }{{a_n-a_{n-1}}!}$.
In other words, the additional observations $X_{n-2},X_{n-3}, ...$ constrict the probability of $X_n$ by adding other known jumps, suggesting that Poisson processes are NOT Markovian, although they are well known to be. What is wrong with this proof?