This is from a proof in my book:
If $\displaystyle A = \{ x : f(x) > 0 \} $, define $\displaystyle A_n = \{ x : f(x) > \frac {1}{n} \} $, then $\displaystyle A = \bigcup ^ \infty _ 1 A_n $
Why is this true?
To show set equality we just need to show inclusion, right? One inclusion is obvious:
Clearly $\displaystyle A_n\subseteq A$ for each $\displaystyle n$ so
$\displaystyle A \supseteq \bigcup^\infty_1 A_n $.
The other inclusion is not much harder. Let $\displaystyle x\in A$. Then $\displaystyle f(x)>0$ so by the Archimedean principle there exists an integer $\displaystyle M$ such that $\displaystyle Mf(x)>1$, and so $\displaystyle f(x)>\dfrac{1}{M}$ thus $\displaystyle x\in A_M\subseteq \bigcup^\infty_1 A_n$. Therefore $\displaystyle A \subseteq \bigcup^\infty_1 A_n $ and so $\displaystyle A = \bigcup^\infty_1 A_n $!