Prove if the set $C=$ {${q \sqrt2 \in \Bbb R | q \in \Bbb Q \cap(0, \sqrt2) }$} is bounded above, bounded below and find its supremum and infimum

Question

Prove if the set $C=\{q \sqrt2 \in \Bbb R | q \in \Bbb Q \cap(0, \sqrt2)\}$ is bounded above, bounded below and find its supremum and infimum. prove it

I know that the infimum is $0$, supremum $=2$. But firstly how can I prove this set is bounded

Does anybody can helpe to prove it?

Answer 1

It seems your only question is how to prove that the set $C$ is bounded.

$$\begin{align} q\in\Bbb Q\cap(0,\sqrt 2) &\implies 0<q<\sqrt 2 \\ &\implies 0\sqrt 2 < q\sqrt 2<\sqrt 2\sqrt 2 \\ &\implies 0 < q\sqrt 2<2 \end{align}$$

Therefore,

$$x\in C\implies 0<x<2$$

So $C$ is bounded by $0$ and $2$.