证明limf(x)(x趋向于x0)=a等价于对任意{xn},当xn趋向于xo时,f(xn)趋向于a.

先证limf(x)(x趋向于x0)=a推岛出对任意{xn},当xn趋向于xo时,f(xn)趋向于a δε
对任意δ,因为limf(x)(x趋向于x0)=a,所以存在ε,当|x-x0|=N时,|xn-x0|再证于对任意{xn},当xn趋向于xo时,f(xn)趋向于a推导出limf(x)(x趋向于x0)=a
用反证法,如果limf(x)(x趋向于x0)=a1,a1不等于a,则由上面的证明对任意{xn},当xn趋向于xo时,f(xn)趋向于a1,与已知矛盾,等其所证
综上limf(x)(x趋向于x0)=a等价于对任意{xn},当xn趋向于xo时,f(xn)趋向于a