Explanation of "Guess and Verify" Analytical Solution To Bellman Equation

Question

According to my textbook, the analytical technique for solving a Bellman's Equation is as follows:

1. Guess a form for $V_0(x)$

2. Solve the maximization problem with respect to the control and obtain a policy function x′=h0(x)

3. Update the guess by plugging the policy function such that $V1(x)=F(x,h0(x))+\beta V0(h0(x))$

4. Repeat the above until $V_{i+1}=V_i$ At this point, the equation is solved.

My question is, why does this technique, specifically the step, $V_{i+1}=V_i $, solve the Bellman equation? Thanks!