Equivalence Relation questions about transitivity

Question

I have a questions that asks the following. Let $S=\mathbb{Z}$ for $a,b \in S$, define the relation $R$ by $aRb \iff ab\ge 0 $ Is $R$ an equilivance relation on $S$

Here is my work so far:

1)Suppose $a \in S$, then $aa \ge 0$ by properties of integers. Hence $aRa$

2) suppose $a,b\in S$ and $aRb$, then $ab \ge 0 $. By properties of integers $ab=ba$, hence bRa

3) Suppose $a,b\in S$ and $aRb, bRc$ then $ab\ge0$ and $bc\ge0$

This final one I do not know what to do for it. I am supposed to show $ac\ge0$ and hence then $aRc$ but I dont know what to do for it.

Answer 1

Let $a=1, b=0$ and $c=-1$. Then, certainly $ab\geq 0$ and $bc\geq 0$ but $ac=-1\not\geq 0$. So $R$ is not transitive.