Considering a finite sequence of real numbers $x_0, x_1, ..., x_n$, which has no particular order, could I write a single statement that says the difference between any two elements of the sequence is less than some real number $y$?

In other words, can I write all of the following inequalities as a single statement?

$|x_0 - x_1| < y, |x_0 - x_2| < y, ..., |x_0 - x_n| < y, |x_1 - x_2| < y, |x_1 - x_3| < y,...,|x_1 - x_n| < y,.., |x_{n-1} - x_n| < y$

EDIT: More specifically, I'm looking for a single inequality, involving $y$ and some linear combination of $x_0, x_1, ..., x_n$, that is only true if the above list of inequalities is true.


Sure, how about "$|x_i-x_j|<y$ for all $i$ and $j$". Or if you want to be more formal; $$(\forall i,j\in\{0,\ldots,n\})(|x_i-x_j|<y).$$ Alternatively, you could write $\max_{i,j}\{|x_i-x_j|\}<y$.

  • $\begingroup$ This made me realize that I'm actually looking for an inequality involving $y$ and some linear combination of $x_0, x_1, ..., x_n$ that has the same meaning as the list of inequalities. I'll update the question now. $\endgroup$ – lthompson Mar 10 at 1:40
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    $\begingroup$ @lthompson I have added an alternative. There is no single inequality of the form $$\left|\sum_{i=0}^nc_ix_i\right|<y,$$ that is equivalent with the above, if that is what you are hoping for. $\endgroup$ – Servaes Mar 10 at 1:48

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