In a simple example, why the set $S$ is unbounded?

Question

• There is a real number $y$ such that $y>\dfrac {1}{1+x^2}$ for any real number $x$. Prove or disprove this question.

Answer. Let $S=\left\{ \dfrac {1}{1+x^2}:x\in\mathbb{R}\right\}$. So since $S=(0,1)\subseteq\mathbb{R}$, then $S$ is unbounded. Hence, there is no such a $y$ in $\mathbb{R}$.

My question is: Why the set $S$ is unbounded? Can you explain? Thanks...

Answer 1

We have $0< \frac{1}{1+x^2} \le 1$ for all $x \in \mathbb R$. Hence, for each $y>1$ we have $y>\frac {1}{1+x^2}$ for all $x \in \mathbb R$.

Answer 2

$$\forall x\in\mathbb R:x^2\ge0\implies x^2+1\ge1\implies\frac1{x^2+1}\le 1\implies 2>\frac1{x^2+1}.$$