  Thus, \(S\) is r.e. just if it can be listed out by some Turing machine. Suppose that \(S\) is r.e. and its elements are enumerated by Turing machine \(M\) as above. We can then describe another Turing machine, \(P\), which, on input \(n\), runs \(M\) in a round-robin fashion on all its possible inputs until eventually \(M\) outputs \(n\). If this happens then \(P\) halts and outputs “1”, i.e., \(P(n)=1\). If \(n \not\in S\), then \(M\) will never output \(n\), so \(P(n)\) will never halt, i.e., \(P(n)=\nearrow\). 