A Problem about Limits of Sequence

I have been asked to solve the following problem:
Let $\set{x_n}_{n=1}^\infty$ be a real sequence defined by $$x_{n+1}=\frac{C}{2}+\frac{x_n^2}{2},$$  with $x_1=C/2$, where $C$ is a constant, try to show that

  1. If $C>1$, then $\set{x_n}_{n=1}^\infty$ is divergent;
  2. If $0< C\leq 1$, then $\set{x_n}_{n=1}^\infty$ is convergent;
  3. If $-3\leq C < 0$, then $\set{x_n}_{n=1}^\infty$ is convergent;

Try to discuss the case of $C< -3$, is $\set{x_n}_{n=1}^\infty$ divergent?

If you have any idea, please tell me! Just leave a word below!


  1. 微积分的助教真悲催啊。《数学分析中的证题方法与难题选解》第8面