I am trying to understand the proof for this theorem (theorem 4.22 of the book 'An introduction to the theory of computation'):
A language is decidable iff it is Turing-recognizable and co-Turing-recognizable.
The first direction is OK. But the other is not, namely:
What I don't understand is why the author assumed that M will halt. We have that A is Turing-recognizable, and thus we are not sure we have a machine that decides it (and thus halts). Same for the complement of A which is also Turing-recognizable.
Thanks in advance