Signature sequences

[Henry in Rotherhithe]

It is hardly worth a formal proof, but essentially it would go:

Definition: For a positive irrational number x, form the numbers y = i +
j*x, where i and j are both positive integers.  Since x is irrational, no
two y values will be the same for different i and j.  Arrange the y values
by size, then the sequence of i values is the signature of x.

Lemma 1.    If i values are the signature for x then j values are the
signature for 1/x.
Proof:      i1+j1*x < i2+j2*x < i3+j3*x iff j1+i1*1/x < j2+i2*1/x <
j3+i3*1/x.

Lemma 2.    First occurences in sig.seq.(x) of integers are the same as
positions of 1 in sig.seq.(1/x).
Proof:      First occurence of n in sig.seq.(x) is when n+1*x appears in
list for x,   equivalent to 1+n*1/x appearing in list for 1/x. Reverse also
true.



My question is why i and j have to be positive in the definition, rather
than non-negative?


> It seems that first occurences in sig.seq.(x) of integers
> is the same as positions of 1 in sig.seq.(1/x).
> With the usual plea for proof,
> ralf