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:

\(\ds 1 - n\)

\(\to\)

\(\ds -\infty\)

\(\ds -1 - \dfrac 1 n\)

\(\to\)

\(\ds -1\)

\(\ds -1 + \dfrac 1 n\)

\(\to\)

\(\ds 1\)

\(\ds 1 + n\)

\(\to\)

\(\ds + \infty\)

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.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text {A}$