# Please explain this proof for me: (arbitrage and bounded set) [closed]

Consider this problem and subsequent proposition:

Part of the proof of this proposition is given here:

Could somebody please explain to me why the existence of the "associative ray" (which I have never heard of before) means there's an arbitrage? I've highlighted that part of the proof in red.

• Where did you get this? – Bob Jansen Apr 23 '18 at 18:57
• I guess that $\pi$ is the price vector and $\theta$ the allocation, but would you mind to define $\pi$, $D$ and $\theta$ for clarity? – Daneel Olivaw Apr 23 '18 at 19:21

• Suppose that $\theta^\star$ os an optimal trading strategy for the agent with optimal consumption allocation of $c^\star$. Now assume there is an arbitrage opportunity $\theta^{arb}$ - i.e. without the investment of any initial endowment it yields a consumption bundle $c^{arb} > 0$. This implies that $\theta^\star + \theta^{arb}$ yields $c^\star + c^{arb} > c^{\star}$. Since $U$ is stricty increasing and $\theta^\star + \theta^{arb}$ satisfies the budget constraint it implies that the initial pair $(\theta^\star, c^\star$) is a solution to the investor's problem.