An infinite sequence
![\,x_{0},x_{1},x_{2},\ldots \,](/media/m/6/d/2/6d2957f7143e8273e7b1deca7b8b8d01.png)
of real numbers is said to be bounded if there is a constant
![\,C\,](/media/m/b/a/a/baadd742c8e5e51598730598d31424c8.png)
such that
![\, \vert x_{i} \vert \leq C\,](/media/m/b/8/7/b871094d4676ed6f06ed4037063fa7d9.png)
for every
![\,i\geq 0](/media/m/1/f/5/1f5266773e4ddea692e2cd866580eea0.png)
. Given any real number
![\,a > 1,\,](/media/m/f/a/1/fa143775b098a8d5e4241c0a2e7bb67b.png)
construct a bounded infinite sequence
![x_{0},x_{1},x_{2},\ldots \,](/media/m/3/4/6/3468f19d4742cb0e2470cd4d95bfc475.png)
such that
for every pair of distinct nonnegative integers
![i, j](/media/m/5/a/a/5aac541b853183ea20751471ec3be677.png)
.
%V0
An infinite sequence $\,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be bounded if there is a constant $\,C\,$ such that $\, \vert x_{i} \vert \leq C\,$ for every $\,i\geq 0$. Given any real number $\,a > 1,\,$ construct a bounded infinite sequence $x_{0},x_{1},x_{2},\ldots \,$ such that
$$\vert x_{i} - x_{j} \vert \vert i - j \vert^{a}\geq 1$$
for every pair of distinct nonnegative integers $i, j$.