1. ## infimum proof question

prove that:

√3 = inf{ x in rationals: x > 0 and x^2 > 3}

2. Let $A=\inf\{x\in \mathbb{Q}:03\}$ and suposse $A\neq \sqrt{3}$. Then $A>3$ or $A<3$.

Now you can use the fact that $\mathbb{Q}\subset \mathbb{R}\subset \overline{\mathbb{Q}}$.

3. Originally Posted by Abu-Khalil
Let $A=\inf\{x\in \mathbb{Q}:03\}$ and suposse $A\neq \sqrt{3}$. Then $A>3$ or $A<3$.

Now you can use the fact that $\mathbb{Q}\subset \mathbb{R}\subset \overline{\mathbb{Q}}$.

What do you mean by $\overline{\mathbb{Q}}$? Because if this is, as usually denoted, the algebraic closure of Q then definitely $\mathbb{R} \nsubseteq\overline{\mathbb{Q}}$. For example, $\pi\notin\overline{\mathbb{Q}}$

Tonio

4. Originally Posted by tonio
What do you mean by $\overline{\mathbb{Q}}$? Because if this is, as usually denoted, the algebraic closure of Q then definitely $\mathbb{R} \nsubseteq\overline{\mathbb{Q}}$. For example, $\pi\notin\overline{\mathbb{Q}}$

Tonio
No, i mean closure in terms of limit points. You usually read as $\mathbb{Q}$ is dense in $\mathbb{R}$.

5. Originally Posted by Abu-Khalil
No, i mean closure in terms of limit points. You usually read as $\mathbb{Q}$ is dense in $\mathbb{R}$.

Oh, I see...but then in fact $\mathbb{R}=\overline{\mathbb{Q}}$

Tonio