Prove that $\sqrt{a_1} + \sqrt{a_2} +\cdots+ \sqrt{a_n}$ is irrational
53660
Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
136403
Prove that $\sqrt{a}\leq\frac{1+a}{2}$
104548
Geometric proof for irrationality of $\pi$
78309
On an expansion of $(1+a+a^2+\cdots+a^n)^2$
58182
How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?
142000
Prove that there is an irrational number and a rational number between any two distinct real numbers
14386
Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$
67605
Different proof that $\sqrt{2}$ is irrational
87111
Prove $\sqrt{2}$ is between $\dfrac{a}{b}$ and $\dfrac{a+2b}{a+b}$
42444
Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$
140949
Proving that for each prime number $p$, the number $\sqrt{p}$ is irrational
124641
Prove that $5^{1/3}+7^{1/2}$ is irrational
149700
Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
37915
$\sqrt{2}$ is irrational proof using well-Ordering Property
139755
Prove that $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational. Generalise this.
146546
Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$
88425
Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$