Question about proving the existence of an arbitrage opportunity

I am having a hard time understanding the reasoning behind a statement in the proof of the following lemma from page 14 (228) of the paper "Martingales and stochastic integrals in the theory of continuous trading" by Harrison and Pliska in 1981.

Lemma: If there exists a self-financing strategy $$\phi$$ (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 the 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 $$t such that $$\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 all values after a given time $$t$$ have 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.

I also suspect that $$\phi_u\cdot S_u\leq 0$$ is a typo and it should be $$\phi_u\cdot S_u\geq 0$$.