Proof (sketch): Suppose L1 and L2 are recognizable. Then there are DFAs A1 = (Q,Σ, T1, q0, F1) and A2 = (P,Σ, T2, p0, F2) such that L1 = L(A1) and L2 = L(A2). We construct A′ such that L(A′ ) = L1 ∩ L2. The idea is to have A′ run A1 and A2 in parallel-keeping track of the state of both machines. It will accept a string, then, iff both machines reach an accepting state on that string.
Let A′ = (Q × P,Σ, T′ , (q0, p0), F1 × F2), where
T′ def= [{((q, pi, (q′, p′), σ) | (q, q′, σi)∈ T1 and (p, p′, σ ∈ T2}.
Then
(You should prove this; it is an easy induction on the structure of w.) It follows then that
