I'm reading this and the first part in Lemma 4 is confusing:
Lemma 4. For all acceptable languages L and L′, the languages L ∪ L′ and L ∩ L′ are also acceptable.
Proof: Let M and M′ be Turing machines that decide L and L′, respectively.
I don't follow how we prove the existence of $M$ and $M'$. We know these languages $L$ and $L'$ are acceptable (as stated) but are they really decidable? From what I understand acceptable language $L$ implies a Turing machine that accepts every string of $w \in L$; a decidable language $L_D$ implies a Turning machine that accepts every string of $w \in L_D$ and halts for every $w \in \Sigma^*$ (or in other words does not diverge for any $w$).