# What can be said about min$\{ord_p(x_n-x_m),ord_p(x_m)\}$, if $ord_p(x_n-x_m)$ can be made big?

More specifically,

Let $(x_n)_{n\geq 1}$ be a sequence of rationals and $p$ be prime.

Prove or disprove: If $ord_p(x_n-x_m)$ can be made as large as desired, then min$\{ord_p(x_n-x_m),ord_p(x_m)\}=ord_p(x_m)$.

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"can be made as large as desired"...I suppose that you mean when $n,m\to\infty$, that is, for all $M\in\mathbb N$, there is $N\in\mathbb N$ such that $ord_p(x_n-x_m)>M$ whenever both $n,m\geq N$. –  Matemáticos Chibchas Jan 31 at 1:15