Suppose we have ${x_{n}}$ as a increasing sequence and another sequence ${y_{n}}$ which is defined as $y_{n}=\frac {x_{n}}{1+|x_{n}|}$ then prove that the sequence $y_{n}$ is increasing and comment upon the fact whether it is cauchy or not !
My problem basically is that in the first part by using the definition of an increasing sequence I am not able to make sense of how to prove that $ x_{n+1} |x_{n}| - x_{n} |x_{n+1}| > 0$ and for the next part i don't have any idea of how to approach the problem ! Please Help
Let $0\leq x \leq y$ . Then $x+x|y|\leq y+y|x|$. If $x \leq y \leq 0$ then $x+x|y|=x-xy\leq y-yx=y+y|x|$. If $x \leq 0 \leq y$ then $x+x|y|=x+xy \leq y-xy= y+y|x|$ because $xy \leq 0$ in this case. This proves that $\frac x {1+|x|} \leq \frac y {1+|y|}$ whenever $x \leq y$. Hence $(y_n)$ is increasing.
The second sequence is increasing and bounded, hence convergent (hence Cauchy). Note that $|y_n| \leq 1$ for all $n$.