I am trying to prove **Problem 5(a)**, in **Section 4** of **Munkres' General Topology**. He gives a hint that says:

Show that given $a\in \mathbb{Z}\_+$, the set $X=\{x| x\in\mathbb{R} \,\,\textrm{and}\,\, a+x\in \mathbb{Z}\_+\}$ is inductive.

I can show that the set $X$ is indeed inductive but how does one go from there to showing that $a+b\in\mathbb{Z}\_+$. It seems like he is asking us to build a set of $b$'s, if I am not wrong, and says that this set is inductive. I think I am missing a logical link.