Could you please help me complete the proof of Case II? Thank you.

Theorem 1 (NAP) implies a > \"r > 01'. Proof. Begin by noticing that if a" > u > til, then the safe asset's payoff per unit invested always exceeds the payoff of the risky asset. This implies that the safe asset dominates the risky asset no investor who prefers more cash (in each state) to less will EVER choose to invest in this risky asset. Likewise, if u > d > r > 0, the risky asset [per unit) dominates the safe asset. Hence, we can conne attention to showing it > '1" > d must hold when the (NAP) obtains. Suppose the condition (NAP) is true, but the conclusion of the theorem is false. This means (NAP) holds, (1 > u and r 2 I] hold, but d 2 r. In this case, there will be an arbitrage portfolio. This contradiction implies the theorem is true. Thus, the main argument is to construct an arbitrage portfolio assuming it > a" Z r. The first step is to nd the possible zero net outlay portfolios. These are combinations of 2: and y such that $80 + an = 0'. This restriction implies there are two possible cases: Case I: a: > 0 and y = mSn/Bg. This is a long position in the stock and a short position in the bond (borrowing money). Case II: 3,! > 0 and 1: = yBu/SO. This is a long position in the bond (lending money} and a short position in the stock