What is convergence analysis, and why is it needed in reinforcement learning?

Convergence analysis is about proving that your policy and/or value function converge to some desired value, which is usually the fixed-point of an operator or an extremum. So it essentially proves that theoretically the algorithm achieves the desired function. Without convergence, we have no guarantees that the value function will be accurate or the policy will be any good, so in other words the proposed RL algorithm can completely fail at serving its purpose even in simple cases.