Let two positive values $x > y$ be given along with an integer $h$. Consider the interval $[0, x]$. I want to split this up into $h$ subintervals each of equal length. This is of course always possible and there's a unique and obvious way of doing it.

But I have two extra requirements. Write the subintervals like $[x_i, x_{i+1}]$. The two requirements are then

  1. The value $y$ must lie on a border of one of these subintervals, i.e. there must be an $i$ for which $x_i = y$.
  2. If we instead had to create twice as many subintervals (so, $2h$), then every border $x_i$ from before is again a border now. So, for example, if in the $h$-subinterval construction, we had $x_i$ as a border, then $x_i$ is again a border in the $2h$-construction.

In order to accomplish this, we have one degree of freedom: $x$. What that means if that if it works for the current value of $x$, then that's what we want, but if it doesn't work, then we are allowed to change $x$ to some other value $x' \approx x$.


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