I have a problem with this task:
Show that this language is recursive enumerable, but not recursive: $L = \{ w \in \{0,1\}^* | M_w(x)\; \text{converges for some input}\; x \}$ (where $M$ is turing machine).
I know how to do it with this one: $L = \{ w\in \{0,1\}^* | w \in L(M)\;\text{for some TM $M$} \} \to$ complement of $L$ is diagonalization, so it is not accepted by any TM, so complement of $L$ is not recursively enumerable and so $L$ is not recursive.
Is there any similar approach for original task, please? Or do you have idea how to proceed?