Question: Let $$S = \{\langle M\rangle\mid \text{TM }M\text{ on input 3 at some point writes symbol “3” on the third cell of its tape} \}.$$ Show that $S$ is r.e. (Turing acceptable) but not recursive (decidable).
I'm bit confused about this question, I'm not even sure how exactly should I start an approach to it. As I study from Sipser's book, in general when we prove a language is undeciable, we use contradiction to show that if $S$ is decidable, then by doing some trick, $A_{\text{TM}}$ is also decidable but it's not. Thus, $S$ is not decidable. However, this question also asked to show $S$ is acceptable. I'm thinking if we show that $S$ is undecidable, it's sufficient to see that $S$ is acceptable?(I'm totally not sure) Any suggestion?