lim(x(n+1)-x(n))/(y(n+1)-yn)存在
設(shè)lim(x(n+1)-x(n))/(y(n+1)-yn)=a
對于任意e>0,存在N使得,對n>N有
|(x(n+1)-x(n))/(y(n+1)-yn)-a|
a-e<(x(n+1)-x(n))/(y(n+1)-yn)
(a-e)(y(N+2)-y(N+1))
(a-e)(y(n+1)-yn)
(a-e)(y(n+1)-y(N+1))
|(x(n+1)-x(N+1))/(y(n+1)-y(N+1))-a|
xn/yn - a=(xn-x(N+1))/(yn-y(N+1))* (yn-y(N+1))/yn+(x(N+1)-a*y(N+1))/yn 湊出上式
|xn/yn - a|<=e|1-y(N+1)/yn|+|(x(N+1)-a*y(N+1))/yn|<=e+|(x(N+1)-a*y(N+1))/yn|
存在N'>N使得對n>N'有
|(x(N+1)-a*y(N+1))/yn|
有|xn/yn - a|<2e
那么lim xn/yn=lim(x(n+1)-x(n))/(y(n+1)-yn)