# Turing recognizable and decidable

Question: Let $$S$$ = $$\{ | TM$$ $$M$$ 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 proof a language is undeciable, we use contradiction to show that if $$S$$ is decidable, then by doing some trick, $$A_{TM}$$ 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?