The title should be almost entirely self explaining, but let me give the specific context and use case I'm thinking about while asking too, it's related to the HadwigerNelson problem and chromatic numbers.
For simplicity take ${\mathbb R}^1$, that is simply a line of real numbers. Now take 2 points at unit distance which are connected, $a=0$ and $a'=1$, then $a$ and $a'$ now strictly need to have different colors. Now add $b$ and $b'$ to the line, these points are the next closest minimum distance points to $a$ and $a'$ (let's assume to their right/in the positive direction). Is their distance...
...strictly 0, which would mean that $a$ and $b$ effectively share the same coordinate. In turn this would also mean that both $a$ and $b$ are connected to both $a'$ and $b'$. Assuming $a$ and $b$ have different colors the coordinate 0 would essentially be duochrome, meaning $a'$ and $b'$ would need different colors than those used by $a$ and $b$. This would mean that coloring ${\mathbb R}^1$ with discrete 0-dimensional objects (points) requires an infinite amount of colors.
or
...an infinitesimal, in which case the coordinate of $b$ would be $0+infinitesimal$. $a$ would strictly only be connected to $a'$, and $b$ only to $b'$. This would mean a 2-coloring of ${\mathbb R}^1$ is possible using only discrete alternating 0-dimensional objects (points).
Your way of thinking suggests that you consider that $\Bbb R$ is well ordered. It is not.
In $\Bbb Z$, the number just after $1$ is $2$. In $\Bbb R$, or even in $\Bbb Q$, the number just after $1$ simply does not exist. Asking what is the nearest number from $1$ is as absurd as asking what is the greatest integer.
Think at a number $1+x$ just a bit greater than $1$. Then the arithmetic mean of this number and $1$ (that is, $1+\frac x2$) is between $1$ and $1+x$. Just like no greatest integer exists because if $n$ is a very big integer, $n+1$ is even bigger.