$x\sim y$ if $|x-y|\le 3$, then is $\sim $ or R an equivalence relation?

Question

Let R or $\sim$ be the relation defined on Z by

$$x\sim y\text{ if } |x-y| \le 3$$ Is $\sim$ an equivalence relation?

It is reflexive and symmetric if I did it correctly. However, I am having doubt about transitive.

Transitive:

Assume $a\sim b$ and $b\sim c$, then, $$|a-b|\le 3\text{ and }|b-c|\le 3$$ $$|a-c|\le |a-b|+|b-c|\le 3+3 =6$$

This is where I am stuck at. I am not sure if I should conclude $|a-c|\le 6$ and it is not transitive.

Answer 1

Take $x=-3,y=0,z=3$. Then, $x \sim y$ since their difference is exactly 3, and $y \sim z$ for the same reason. But $x \not\sim z$ since their difference is exactly 6.

Answer 2

Think about pairs $0,3$ and $3,6$ -- is $(0,6) \in R$?

Answer 3

We have $6\sim3$ and $3\sim0$, but is $6\sim0$ true?