Question

Exercise 3. Let α and β be two real numbers so that α ≤ β and β ≤ α. Then prove α = β.

Answer 1

$\alpha \leqslant \beta$ $\implies$$\alpha \isin (-\infty,\beta]$

$\beta \leqslant \alpha$ $\implies$ $\alpha\isin [\beta, \infty)$

$\implies \alpha \isin (-\infty,\beta] \bigcap [\beta,\infty)=[\beta\beta]=\{\beta\}$

$\implies \alpha=\beta$