I have this lemma in front of me an dI have hard time understanding the reasoning behind a statement in the proof.
Lemma: If there exists a self financing stragegy $\phi$ (not necessarily admissible) with $V_{0}(\phi)=0, V_{T}(\phi)\geq 0$ and $\mathbb{E}[V_{T}(\phi)]>0 $ then there exists an arbitrage opportunity.
Here admissible stands for $V(\phi)\geq 0$, $\phi,S\in\mathbb{R}^n $ and $V_t(\phi)=\phi_t\cdot S_t$ which is a scalar product.
Proof: If $V(\phi)\geq 0$ then $\phi$ is admissible and hence is an arbitrage opportunity itself, and we are done. Otherwise there must exist $t<T,A\in\mathcal{F_t}$ such that $\phi_t\cdot S_t=a<0$ on $A$ and $\phi_u\cdot S_u\leq 0$ on $A$ for all $u>t$.
Here $\phi_t\in\mathcal{F_{t-1}}$ and $ S_t \in \mathcal{F_{t}}$
What I don't understand is why does all values after a given time $t$ has to be non positive. My intuition tells me that I should have some which are non positive, but not necessarily consecutively. And also the statement contradicts $ V_{T}(\phi)\geq 0$ in my opinion. With this I want to create a new strategy which is both self financing and admissible. The problem with taking $t$ non consecutive is that the new strategy doesn't satisfy either the admissibility condition or the self financing condition.
Hope the question is clear.