📝 题目
例 15 设 ${x}_{n} > 0$ ,求证: $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = 0 \Leftrightarrow \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{{x}_{n}} = + \infty}$ .
💡 答案与解析
证 “ $\Rightarrow$ ” $\forall M > 0$ ,对 $\varepsilon = \frac{1}{M} > 0$ ,因为 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = 0}$ ,所以 $\exists N$ ,
当 $n > N$ 时,有
$$ 0 < {x}_{n} < \varepsilon \Rightarrow \frac{1}{{x}_{n}} > \frac{1}{\varepsilon } = M,\;\text{ 即 }\;\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{{x}_{n}} = + \infty . $$
“ $\Leftarrow$ ” $\forall \varepsilon > 0$ ,对 $M = \frac{1}{\varepsilon } > 0$ ,因为 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{{x}_{n}} = + \infty}$ ,所以 $\exists N$ , 当 $n > N$ 时,有
$$ \frac{1}{{x}_{n}} > M \Rightarrow {x}_{n} < \frac{1}{M} = \varepsilon ,\;\text{ 即 }\;\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = 0. $$