Assume there is a multi period discrete market i.e. a finite Probability space, finite time periods $\lbrace 0,1\ldots, n\rbrace$, a stochastic process $(S(t))_{t\in\lbrace 0,1,\ldots,n\rbrace}\subseteq \mathbb{R}^d$ with $S_0(t)>0$ for all $t\in\lbrace 0,\ldots,n\rbrace$, that is adapted to the Filtration $\mathscr{F}_t:=\sigma(S(k), k\leq t)$ ($t\neq 0, n$), $\mathscr{F}_n:=2^{\Omega}$, $\mathscr{F}_0:=\lbrace 0,\Omega\rbrace$.
How can I prove:
If there exists some $t\in \lbrace 1,\ldots, n \rbrace$ such that $P(\phi(t)^T S(t)<0)>0$ for all predictable, selffinancing trading strategies $(\phi(t))_{t\in\lbrace 1,\ldots, n\rbrace}\subseteq\mathbb{R}^d$, i.e. $\phi(t)$ is $\mathscr{F}_{t-1}$ measurable and satisfies $\phi(t)^T S(t)= \phi(t+1)^TS(t)$ for all $t\in \lbrace 1, \ldots, n-1\rbrace$, then
there can't be an arbitrage possibility in this market.
Where an arbitrage opportunity means:
There is a self financing trading strategy $\psi(t)$ such that
$\psi(1)^T S(0)=0$ and $P(\psi(n)^T S(n)\geq 0)=1$ and $P(\psi(n)^T S(n)>0)>0$