Let $f:\left]-1,\infty\right[ \to \mathbb{R}$ be defined by $$f(x) := \frac{1}{1+x \cdot |x|} .$$ I want to prove that $f$ is not bounded above.
Here is my attempt: I assume that $f$ is upper bound by $k$:
$$\exists k \gt 0 : f(x) \le k \qquad x \in \left]-1, \infty\right[ $$
$$\Rightarrow \qquad \frac{1}{1+x\cdot\left|x\right|} \le k .$$
But at this point I don't know how to proceed. I can't find the contradiction.