In example 1.1 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition, the author shows that there is no rational number $p$ such that $p^2 = 2$; moreover, the set $A$ of all rational numbers $q$ such that $q^2 < 2$ has no largest element in the set of rationals and the set of all rational numbers $r$ such that $r^2 > 2$ has no smallest element.
Now we would like to state the following.
Let $n$ be a positive integer greater than $1$. Then there is no rational number $p$ such that $p^n = 2$; moreover the set $A$ of all rationals $q$ such that $q^n < 2$ has no largest element, and the set $B$ of all rationals $r$ such that $r^n > 2$ has no smallest element.
How to prove this statement, using the same idea as Rudin has used?
How can we state and prove an even more generalised version of the above statement, with $2$ replaced by an arbitrary (positive) integer---or perhaps even by an arbitrary rational number?