Union of Family/Examples/Size of y-1 lt n and Size of y+1 gt 1 over n

Example of Union of Family
Let $I$ be the indexing set $I = \set {1, 2, 3, \ldots}$

Let $\family {T_n}$ be the indexed family of subsets 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:
 * $\ds \bigcup_{n \mathop \in I} T_n = \R \setminus \set {-1}$

Proof
From Size of y-1 lt n and Size of y+1 gt 1 over n:


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

We have that:


 * $\paren {\openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n} } \subseteq \paren {\openint {1 - \paren {n + 1} } {-1 - \dfrac 1 {n + 1} } \cup \openint {-1 + \dfrac 1 {n + 1} } {1 + \paren {n + 1} } }$

That is:
 * $T_n \subseteq T_{n + 1}$

and so:
 * $\ds \bigcup_{n \mathop \in I} T_n = \lim_{n \mathop \to \infty} \openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n}$

As $n \to \infty$, we have:

and it follows that:
 * $\lim_{n \mathop \to \infty} \openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n} = \openint \gets {-1} \cup \openint {-1} \to$

whence the result.