Is there a strictly-decreasing sequence $(x_n)$ with $x_n \rightarrow 0$ and $\frac{x_{n+1}}{x_n}\rightarrow1$?

Question

Does there exists a strictly decreasing sequence $(x_n)_{n \in \mathbb{N}}$ such that $x_n \rightarrow 0$ and $\frac{x_{n+1}}{x_n}\rightarrow1$ ?

Answer 1

Let consider

$$x_n=\frac 1 n$$

or more in general for $a>0$

$$x_n=\frac 1 {n^a}$$

Answer 2

$X_n$=$1/n$ should come to your mind first then raise any positive power to $n$, the result still holds.