📝 题目
例 9 设 ${x}_{n} \leq a \leq {y}_{n}\left( {n = 1,2,\cdots }\right)$ ,且 $\mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{x}_{n} - {y}_{n}}\right) = 0$ . 求证:
$$ \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = \mathop{\lim }\limits_{{n \rightarrow \infty }}{y}_{n} = a. $$
💡 答案与解析
证 ${x}_{n} \leq a \leq {y}_{n} \Rightarrow 0 \leq a - {x}_{n} \leq {y}_{n} - {x}_{n}\underset{0 \leq a - {x}_{n} \leq {y}_{n} - {x}_{n}}{\underbrace{\mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{y}_{n} - {x}_{n}}\right) = 0}}\underset{n \rightarrow \infty }{\underbrace{\mathop{\lim }\limits_{{n \rightarrow \infty }}{y}_{n} = a}}$ .
\subsubsection{三、用单调有界定理}