Allow me to elaborate further on the “hint” which appears in the description of problem 2 on problem set 4. There it says the following:
Consider a portfolio that is long one option with strike price K1, long one option with strike price K3, and short two options with strike price K2.
The key here is to enumerate the payoffs generated by this portfolio when the terminal share price is 1) less than K1, 2) between K1 and K1, 3) between K2 and K3, and 4) greater than K3. When you do this, you will find that the value of this portfolio is always non-negative at the expiration date; thus, in the absence of arbitrage possibilities, the current market value of such a portfolio must also be non-negative. The same logic applies to problem 3.