Such a lower bound does not exist in general.

E.g., for natural $j$ and natural $n\in((j-1)!,j!]$, let $x_n:=1/(j-1)!$, with $x_1:=1$, so that $x_n>1/n$ for all natural $n$. Also, for any fixed natural $k\ge2$ and all natural $j\ge k$, we have $kj!\in(j!,(j+1)!]$ and hence $$\frac{x_{kj!}}{x_{j!}}=\frac{(j-1)!}{j!}\to0$$ as $j\to\infty$, so that $$\inf_{n\ge1}\frac{x_{kn}}{x_n}=0.$$

Such a lower bound does not exist in general.

E.g., for natural $j$ and natural $n\in((j-1)!,j!]$, let $x_n:=1/(j-1)!$, with $x_1:=1$, so that $x_n\ge1/n$ for all natural $n$. Also, for any fixed natural $k\ge2$ and all natural $j\ge k$, we have $kj!\in(j!,(j+1)!]$ and hence $$\frac{x_{kj!}}{x_{j!}}=\frac{(j-1)!}{j!}\to0$$ as $j\to\infty$, so that $$\inf_{n\ge1}\frac{x_{kn}}{x_n}=0.$$