What are the conditions for the convergence of SARSA to the optimal value function? Is it correct that for SARSA to converge to the optimal value function (and policy)


The learning rate parameter  $\alpha$ must satisfy the conditions:
$$\sum \alpha_{n^k(s,a)} =\infty \quad \text{and}\quad \sum \alpha_{n^k(s,a)}^{2} &lt;\infty \quad \forall s \in \mathcal{S}$$
where $n_k(s,a)$ denotes the $k^\text{th}$ time $(s,a)$ is visited
$\epsilon$ (of the $\epsilon$-greedy policy) must be decayed so that the policy converges to a greedy policy.
Every state-action pair is visited infinitely many times.


Are any of these conditions redundant?
I have the conditions for convergence in these notes SARSA convergence by Nahum Shimkin.


The Robbins-Monro conditions above hold for $α_t$.
Every state-action pair is visited infinitely often
The policy is greedy with respect to the policy derived from $Q$ in the limit
The controlled Markov chain is communicating: every state can be reached from any other with positive probability (under some policy).
$\operatorname{Var}{R(s, a)} &lt; \infty$, where $R$ is the reward function