proving existence of TM that accepts the next language

Question

I have an idea of how to approach the problem, but I'm not sure about it. Given a Turing Machine, I can check how many states the machine has, and somehow by the number of states to know if the run uses endless cells of the tape. However, I am not sure how I can use the number of states and what I should check. What confuses me is that entering an endless loop does not necessarily require the use of endless cells of the film.

$L_k=\{<M>| \,M\,\,\,is\,\,a\,\,TM\,\,that\,\,uses\,\,at\,\,most\,\,k\,\,tape\,\,cells\,\,\,when\,\,\,running\,\,\,on\,\,\,\epsilon\}$

prove that: $L^*=\bigcup\limits_{k\,\in\,\mathbb{N}} L_k\,\in\,\,\,RE\setminus R$

Answer 1

The language $L^*$ consists of all Turing machines $M$ which either eventually halt or repeat a configuration. For each machine $M$, by simulating the machine you can easily observe that one of these possibilities has happened.