Playing around with two frequently asked questions (Stopping times of càdlàg processes, Stopping times and the left limit) I got curious if the concept of preservable stopping times (see the first answer to the second question) couldn't be used for a simple proof of the following implication:
$$(M_t)_{t\in [0,T|} \mbox{ càdlàg }, P(M_T>0=1) \Rightarrow P( \inf\limits_{t\in [0,T]} M_t>0)=1$$
So define $\tau$ as in the first question for $A:=[-\infty, 0]$, so $$\tau := \inf\{ t\geq 0: X_t\in A \mbox{ or } X_{t-}\in A\}.$$
Define for $n\in\mathbb{N}$ the stopping times: $$\tau_n := \inf \left\{ t\geq0: M_t < 1/n \right\}.$$
It is obvious, that $\tau_n\leq \tau$ a.s. But does the fact $\tau_n<\tau$ also hold almust surely on $\{\tau<T\}$? (I think it is not needed on $\{\tau =T\}$, because we know, that $M$ does not jump to 0 at $T$.) One could imagine, that $M$ stays above $1/n>x>2/n$ for a time and then jumps from $x$ to $0$ and so we would have $\tau_1<\tau_2=\tau$. In this case, the Theorem stated in the second question would not be useful. But I think, we can proof that $\tau_n<\tau$ holds on $\{ 0< \tau <T\}$ ( even on $\{ \tau <T\}$, because of $E[M_T]=E[M_0]$ we can not have $M_0\leq 0$, as it would be a contradiction to $P(M_T>0)=1$).
So let's assume, that $\{\tau_n=\tau<T\}$ has positive probability.That means $ P(\{M_{\tau_n} \leq 0\})>0$. Because of $$E[M_T| M_{\tau_n} ]= M_{\tau_n}$$ we get $P( M_T \leq 0)>0$, right? That would be a contradiction.
So we really have $\tau_n<\tau$ for all $n\in\mathbb{N}$ and could apply the Theorem stated in https://math.stackexchange.com/a/3402907/615367 ?
But I think this proof could be shortend without this concept: So assuming $P( \inf\limits_{t\in [0,T]} M_t\leq 0)>0$ and $P( M_T>0)=1$ we get $$P( \inf\limits_{t\in [0,T)} M_t\leq 0)>0$$ and so $$P\left( \inf\left\{ t\geq 0: M_t \in A \mbox{ oder }M_{t-}\in A \right\} <T\right)=P( \tau <T ) >0$$
Because of $$ E\left[ M_T | M_{\tau_n } \right]= M_{\tau_n }$$ and $\{ \tau_n < T \} \subset \{ M_{\tau_n } \leq 0 \}$ we immediately get $P\left( M_t \leq 0 \right)>0$ which is again a contradiction?
So am I assuming something terribly wrong here or did I play around with these concepts correctly? I am not quite sure at the moment, that there isn't something wrong.