My question is regarding the Bellman equation regarding strategy $\sigma^{(1)}$ on the last 2 lines (I have attached pictures of the book below). If we know that all future states will have value of 0, since we have spent a=s in period 1, why do we need to keep this part $\delta W\sqrt{\sigma^{(1)}(s-a)} = \delta \sqrt{(s-a)}$ in the expression?
Here is the reading relating to the question ---
This is policy iteration solution algorithm in which we iterate on the policy (strategy) function $\sigma^{(j)}$. The initial guess is $\sigma^{(0)}(s)=0$. The next iteration policy $\sigma^{(1)}$ is found by solving the optimization problem in the Bellman equation using the continuation value resulting from $\sigma^{(0)}$, that is $W(\sigma^{(0)})(s)=0$. Similarly, on the next iteration, $\sigma^{(2)}$ policy is found by soling the Bellman equation with the continuation value $W(\sigma^{(1)})(s)=\sqrt{s}$. Note however that in Bellman equation $W(\sigma^{(1)})$ is calculated at $s-a$.