Proving the transitivity of $a\sim b$, where $a\sim b$ iff $ab \leq a \lvert b\rvert$

Question

I would like to know how to prove the transitive relation for $\sim$ according to the following definition:

Suppose $\sim$ is defined on the set of the integers as follows : $a\sim b$ iff $ab \leq a \lvert b\rvert$

How do I do this? Please somebody explain to me.

Answer 1

The relation is not transitive.

Evidently we have for all integers $a,b$:$$a\sim0\wedge0\sim b$$

So if $\sim$ is indeed transitive then $a\sim b$ must be true for every pair of integers $a,b$.

This is not the case.

Find integers $a,b$ yourself for which $a\sim b$ is not true.