Supermartingale $(10$ points$)$

Assume asset price $S_k,k=0,1,$cdots,$N$ follows an $N-$period binomial model, the hedging

process $x_{k+1}$ for the American put option is

$x_{k+1}={}_k{S}_{k+1}+(1+r)(x_k-c_k-{}_k{S}_k)$

where ${}_k$ is the hedging ratio and $c_k$ is the amount consumed at time $k($Please refer to the

lecture notes for the details of explanation of hedging process$).$ Show that the discounted

value of $x_{k+1}$ is a supermartingale under the risk$-$neutral probability measure.