Inferring polynomial rate of convergence from polynomial bound

Question

Let $x_n$ be a non-negative valued sequence and suppose that the following hold:

• $\lim\limits_{n\to\infty} x_n =0$
• There exists some polynomial function $p$ of degree at-least $1$ such that: $$ \|x_n\| \leq p(n^{-1}); $$ (not that there is no assumption that $p(0)=0$!)
• $x_n\neq 0$ for any $n>0$.

Can we deduce the existence of a rational function $q$ and some positive integer $N$ satosfying:

1. $q(0)=0$
2. $\|x_n\|\leq q(n^{-1})$ for all $n\geq N$.

Answer 1

The answer is no (assuming that $\|x\|:=x$ for real $x\ge0$). Indeed, as noted by mathworker21, your condition on the existence of $p$ is always satisfied, by choosing, e.g., $p(t)=c+t$ with $c:=\sup_n x_n$.

On the other hand, your condition on the existence of $q$ will not hold e.g. if $x_n=1/\sqrt n$ for all natural $n$. Indeed, suppose the contrary: that there is some rational function $q$ such that $q(0)=0$ and $1/\sqrt n\le q(1/n)$ for all large enough $n$. Then $$q(t)=t^k\frac{P(t)}{R(t)}$$ for some integer $k\ge1$, some polynomials $P$ and $R$ with $R(0)\ne0$, and all $t$ in a neighborhood of $0$. So, $q(1/n)=O(1/n^k)=o(1/\sqrt n)$, which contradicts the condition that $1/\sqrt n\le q(1/n)$ for all large enough $n$.