Arbitrage pricing for financial options

An arbitrage exists if there is a portfolio of options with initial value $V_0 \leqslant 0$ , that is with a nonpositive cost of construction, such that upon expiration at time $T$ we have

$$\tag{*}\mathbb{P}(V_T \geqslant 0) =1, \quad \mathbb{P}(V_T > 0) > 0. $$

If $C(K_1) < C(K_2)$, then a portfolio that leads to arbitrage is long one option with strike $K_1$ and short one option with strike $K_2$. The initial cost here is $C(K_1) - C(K_2) < 0$. Using the payoff scenarios that you correctly specified, show that the arbitrage condition (*) is met. This requires that the probability of the stock price exceeding $K_1$ be greater than zero -- which is true under any reasonable stochastic model for the price of a common stock.

Note also that $C(K_1) = C(K_2)$ also leads to an arbitrage as the initial cost of the long-short portfolio is zero and you can make money from nothing with non-zero probability. To preclude arbitrage we must see $C(K_1) > C(K_2)$.

All of this assumes that you can transact at those prices, i.e., there is no bid-ask spread.