# Difference of any two elements of a sequence being less than some number?

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 "$$|x_i-x_j| for all $$i$$ and $$j$$". Or if you want to be more formal; $$(\forall i,j\in\{0,\ldots,n\})(|x_i-x_j| Alternatively, you could write $$\max_{i,j}\{|x_i-x_j|\}.
• This made me realize that I'm actually looking for an inequality involving $y$ and some linear combination of $x_0, x_1, ..., x_n$ that has the same meaning as the list of inequalities. I'll update the question now. – on-pasta Mar 10 '19 at 1:40
• @lthompson I have added an alternative. There is no single inequality of the form $$\left|\sum_{i=0}^nc_ix_i\right|<y,$$ that is equivalent with the above, if that is what you are hoping for. – Servaes Mar 10 '19 at 1:48