Size of y-1 lt n and Size of y+1 gt 1 over n

Theorem
Let $T_n \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:
 * $T_n = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}$

Then:
 * $T_n = \openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n}$

Proof

 * Union-of-Family-Example-1.png

First note that:

We have: