Let, $S\subset \mathbb{Q}$ defined by $S =\{ x\in\mathbb{Q} : x>0, x^2<2\}$

Question

Let, $S$ be a subset of $\mathbb{Q}$ defined by $S = \{ x \in\mathbb{Q} : x>0, x^2<2\}$ Show that $S$ is a non -empty subset of $\mathbb{Q}$ and bounded above but $\sup S$ does not belong to $\mathbb{Q}$.

Answer 1

I am a clairvoyant:

$$S=\{x \in \mathbb Q: x>0 , x^2 <2 \}.$$

It should be clear, that $1 \in S$.

Now show that $0<x <2$ for all $x \in S$.

Then show that $\sup S= \sqrt{2}.$