函数极限的柯西收敛准则

以下内容来自中科大数学分析教程P73,定理2.4.7
\(函数在x_{0}点的极限的定义\)
\(若存在l，\forall \epsilon>0，\exists\delta>0，使得当|x-x_{0}|<\delta\)
\(则有|f(x)-l|<\epsilon,即称l为f(x)当x趋近于x_{0}的极限\)

\(定理：函数f(x)在x_{0}处有极限的充要条件是\forall \epsilon>0,\exists\delta>0,\)
\(\quad\quad 使得任意x_{1},x_{2}\in U(x_{0},\delta)时，有\)
\(\quad\quad |f(x_{1})-f(x_{2})|<\epsilon\)
证明：
1.必要性
\(若f(x)在x_{0}点的极限为l，即\forall \frac{\epsilon}{2}>0,\exists\delta,当x_{1},x_{2}\in U(x_{0},\delta)\)
\(有|f(x_{1})-l|<\frac{\epsilon}{2},|f(x_{2})-l|<\frac{\epsilon}{2}\)
\(则：|f(x_{1})-f(x_{2})|=|f(x_{1})+l-l-f(x_{2})|\)
\(\quad\quad \leqslant |f(x_{1})-l|+|f(x_{2})-l|\)
\(\quad\quad\leqslant\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon\)