How can I show that $\sqrt[2]{2+\sqrt[3]{3+\sqrt[4]{4+\cdots}}} $ (repeated $n$ times) is irrational using induction?
I know the base case for $n=1$ looks like: $\sqrt[2]{2}$ is irrational.
I also know that I need to assume it is true for an arbitrary $n$ and show that it is true for $n+1$. That is where I get stuck.
Can I use the fact that $\sqrt[n]{n}$ is irrational, $n\geq 2$?
After proving that $n^{1/n}$ is irrational, prove the following stronger statement by induction on $r$:
If $n_1, n_2, \dots, n_r$ are positive integers, $a_1, a_2, \dots, a_r$ are positive rational numbers, and $x$ is a positive irrational number, then the number $$\large \sqrt[n_r]{a_r+\sqrt[n_{r-1}]{a_{r-1}+\dots \sqrt[n_1]{a_1 + x}}} $$ is irrational.