Convergence of Sarsa($\lambda$)

Is there any theorem on the convergence of the Sarsa($\lambda$) Algorithm?

I am currently working through the theory of Reinforcement Learning with the lecture by David Silver and the book of Sutton Barto. I could not find an answer to my question. I found comments and theorems on the convergence of TD($\lambda$) and one-step Sarsa, but nothing for Sarsa($\lambda$). Same result after hitting Google for an hour. Does the convergence follows naturally as Sarsa($\lambda$) is basically the equivalent of TD($\lambda$) in the control setting?

Any help appreciated.