Let $A$ and $B$ be $\mathbb{Z}_2$-graded algebras, i.e.
$A=A_0 \oplus A_1$,
$B= B_0 \oplus B_1$.
I am trying to show that the graded tensor product $A \otimes B = (A \otimes B)_0 \oplus (A \otimes B)_1$;
$(A \otimes B)_0 = (A_0 \otimes B_0) \oplus (A_1 \otimes B_1)$,
$(A \otimes B)_1 = (A_0 \otimes B_1) \oplus (A_1 \otimes B_0)$,
equipped with multiplication $(a \otimes b)(a' \otimes b')=(-1)^{\text{deg}(a')\text{deg}(b)}aa' \otimes bb'$ is also a $\mathbb{Z}_2$-graded algebra.
I can show bilinearity and associativity, but how do I show that $(A \otimes B)_i (A \otimes B)_j \subseteq (A \otimes B)_{i+j}$?
For instance, I want to show that $(A \otimes B)_0 (A \otimes B)_0 \subseteq (A \otimes B)_0$, but I don't know how to proceed.