Prove that the price of the European put option is a convex function of the strike price in one-step binomial model.
In other words, if $P_E(X)$ is the price of the European put option in one-step binomial model, prove that for each $\alpha ∈ (0,1)$ the following inequality holds:
$$P_E(\alpha X+(1-\alpha)X') \le \alpha P_E(X)+(1-\alpha)P_E(X')$$
The right side of the inequality looks like an expectation, and the left side just looks like you can distribute the $P_E$ to make it look like the right side but that doesn't make sense because $P_E$ should depend on $X$.
I don't understand exactly what I am supposed to be doing here?
You are trying to compare a payoff of the option with the current price with the payoff of 2 options with up and down prices, X and X'. The latter is higher (for both calls and puts) because your positive payoff kicks in for a wider range (there is more chance that either X or X' will be in the money). If you draw a graph of a simple payoff in both cases it becomes obvious.