Supposing $x,y \in \mathbb{Q}^+$, does there exists $n \in \mathbb{Z}^+$ such that $x^n>y$ whenever $x>1$, or $x^n<y$ whenever $0<x<1$?

Question

If $x,y$ are rationals and both positive numbers, how can I show:

1. if $x>1$ there is a positive interger $n$ such that $x^n>y$,
2. if $0 < x < 1$ there is a positive interger $n$ such that $x^n<y$.

I'd particularly like a proof that uses the binomial theorem.

Answer 1

1. Write $x=(1+t)$, where $t>0$. Then $$x^n=(1+t)^n=\sum_k \binom nkt^k = 1+nt+\binom n2t^2+\dots \ge 1+nt$$ (so far this is called the Bernoulli inequation). In order that this be $>y$, try to find a corresponding $n\in\Bbb N$.

2. The easiest if we reduce it to 1. by considering $1/x$ which is $>1$ in this case, and $1/y$ in place of ($y$ in 1).