Let $M_t$ an adapted, a.s. bounded process taking values on the integers.
So it is a continuous time prcess with discrete values.
Define: $P_t^i=\mathbb{E}[M_{\tau_i}|\mathcal{F}_t]$
Where stopping times $\tau_{1}$,...,$\tau_{n}$ represent times where the process changes value, i.e.
$\tau_{1} = \inf \{ u>t|M_u-M_{u^{-}} \neq 0\}$
and
$\tau_{i+1} = \inf \{u>\tau_i|M_u-M_{u^{-}} \neq 0 \}$
Consequently, the processs $P_t^i$ is a martingale up to the stopping time $\tau_i$ as it is a constant up to that stopping time (is this rigorous by the way?)
Now suppose the limit
$P_t = \lim_{i\to\infty}{P_t^i}$
exists. How can we prove it is a martingale?