Some notation: We consider a financial market with $d+1$ assets, the $0$-th asset is considered the risk-free asset, the others are the risky ones. The vector $\overline \pi \in \mathbb R^{d+1}$ denotes the pricing vectors, and the random vector $\overline S = (S^0,S^1,\ldots, S^d)$ denotes the vector of the random variables corresponding to the assets. Further we have $\pi^0 = 1, S^0 \equiv 1 + r$, as these are risk-free.
Here a portfolio $\overline \xi \in \mathbb R^{d+1}$ is called an arbitrage opportunity if $\overline \xi \cdot \overline \pi \le 0$ but $\overline \xi \cdot \overline S \ge 0$ P-a.s. and $P(\overline \xi \cdot \overline S > 0) > 0$. And the market model is called non-redundant if $$ \overline \xi \cdot \overline S = 0 ~\mbox{P-a.s.} \Rightarrow \overline \xi = 0 $$ i.e. non of the assets $S^i$ could be synthezied by investing in the others.
Show that in a non-redundant and arbitrage-free market model the set $$ \{ \overline \xi \in \mathbb R^{d+1} : | \overline \pi \cdot \overline \xi = w \mbox{ and } \overline \xi \cdot \overline S \ge 0 ~ \mbox{P-a.s.} \} $$ is compact for any $w > 0$.
First, as this is a subset of the euclidean space $\mathbb R^{d+1}$, compactness is equivalent with bounded and closed. But already showing boundedness gives me a hard time, I do not see where to start?
Also, I guess a non-redundant market model also implies that it is arbitrage-free, so the presupposition of just arbitrage-free'ness should be enough.