请教一道数列极限的问题：设数列{Xn}是正数列,且Xn+1/Xn→1/2(n→∞),试证明：数列{Xn}从某一项起一定单调减少

Question

请教一道数列极限的问题：
设数列{Xn}是正数列,且Xn+1/Xn→1/2(n→∞),试证明：数列{Xn}从某一项起一定单调减少

Answer 1

用极限定义证明：（已知）对任意正数 eps >0,存在N,当n>N时 |Xn+1/Xn - 1/2| < eps所以有了Xn+1/Xn - 1/2< epsXn+1/Xn < 1/2 + epsXn+1< 1/2 Xn + eps * Xn所以Xn+1 - Xn < -1/2 Xn + eps * Xn = (-1/2 +eps) *Xn只...