prove the condition $D<R<U$ is equivalent to the absence of arbitrage:
R = risk free investment rate of return. U and D are returns corresponding to the upward/downward price movements of a risky security.
The basic idea of arbitrage is to buy low, sell high with no risk or initial investment
So Assume R >= U or R<=D (contrapositive)
At time 0
1) We short the security with price S (get +S)
2) We invest the security into the risk free investment (-S) Balance at time 0 = 0
At time 1
1) We get our risk free investment back (+S(1+R))
2) We buy back one share of the stock. There are two scenarios
a) Stock price goes up: (-S(1+U)) Profit = S(1+R)-S(1+U) >= 0 because R>=U (profit is always non negative, there is arbitrage)
b) Stock price goes down: (-S(1+D)) Profit = S(1+R)-S(1+D) <= 0 because R<=D (from assumption), but this shows profit can be negative here-there is no sure risk free profit arbitrage! so i'm stuck here.