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，existsdelta>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,existsdelta>0,)
(quadquad 使得任意x_{1},x_{2}in U(x_{0},delta)时，有)
(quadquad |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,existsdelta,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})|)
(quadquad leqslant |f(x_{1})-l|+|f(x_{2})-l|)
(quadquadleqslantfrac{epsilon}{2}+frac{epsilon}{2}=epsilon)