Is the following relation transitive: $ (x,y) \in R \iff x-y \in \mathbb{Z}$?

Question

Question:

Define the following relation on real numbers:

$ (x,y) \in R \iff x-y \in \mathbb{Z}$

Is the following relation transitive?

My attempt:

If $(a,b) \in R $ and $ (b,c) \in R$ then $ a-b \in \mathbb{Z}$ and $ b -c \in \mathbb{Z}$. How can I prove that $ a -c \in \mathbb{Z}$?

Answer 1

The sum of two integers is again an integer.