Given random variables $x_1,x_2,\ldots,x_n$ which are i.i.d. with $x_i \sim \mathcal N(\eta, \sigma^2), \forall 1 \leq i \leq n$.
I tried to find the probability that none of these random variables are separated by a value less than $d$ such that
$$\Pr(x_k-x_m \geq d)$$
for any $k$ and $m \in [1,n]$. I have looked for some questions here in Stackexchange for finding the difference between two random variables $X$ and $Y$ but I don't know how that scale for a set of $n$ random variables so as $n$ increases the above probability will decrease.
I am asking for guidance here to solve it.