Ring of rational integers is Euclidean with norm equal to binary logarithm of absolute value

Statement
The ring of rational integers $$\mathbb{Z}$$ is a Euclidean domain with Euclidean norm given by:

$$N(a) = [\log_2 |a|]$$.

where $$[x]$$ denotes the greatest integer less than or equal to $$x$$.

Further, this is the slowest growing norm possible.

Related facts

 * Ring of rational integers is Euclidean with norm equal to absolute value
 * Ring of rational integers is not uniquely Euclidean for any norm

Proof idea
The key idea here is to show that the norm of half an integer is one less than the norm of that integer, and that we can always choose a remainder whose absolute value is less than or equal to half the divisor. This can be done because we have a choice of whether to pick a positive or a negative remainder.