To see this, let us first note that the only *x* values that need to be
considered are those in 0<*x*<1, because by adding or subtracting
whole integers to *x* one doesn't change which elements get rounded up
and which get rounded down.

Next, we only really need to analyse *x* values in 0<*x*<1/2,
because by switching *x* for 1-*x*, every element that was previously
rounded up will now be rounded down and vice versa.

In our analysis we will now use the function "modulo" extensively, so, for
clarity, let us define it. For two positive real numbers, *a* and *b*,
the value *c*=*a* mod *b* is the unique real number *c* in
the
range 0≤*c*<*b* such that if you subtract it from *a* the
result is a multiple of *b*, in the sense that
*k*=(*a*-*c*)/*b* is an integer. This integer, *k*, is
referred to as *a* div *b*.

We note that we can equally consider all elements in our sequence modulo 1, for this does not change their rounding properties.

Let *y* be 1/2 mod *x*, and let us divide the range (0,1), where
all elements lie, to the following sub-ranges:

*D*=(0,1/2-*y*]*M*=(1/2-*y*,1/2+*y*)*U*=[1/2+*y*,1)

Consider now the elements of the sequence modulo 1. The sequence begins
with exactly *k*_{0} elements in *D*. These are all rounded
down. Then, there may or may not be an element in *M*. After that, there
are exactly *k*_{0} elements in *U*, which are all rounded
up.
Then the cycle begins again. The only difference from cycle to cycle is whether
or not there is an element inside *M*, and whether that element is
rounded up or down. The *M* element is at most unique at each cycle.
(However, as *n* approaches infinity, so does the total number of elements
that fall into the *M* region.)

Let us divide the original sequence into two sub-sequences. The subsequence
*H*_{0} will be composed of all elements in *D* and *U*,
whereas the *T*_{0} subsequence will be composed of all elements
in *M*.

For an *x* value in (0,1/2), in the subsequence *H*_{0}, the
value of *d*-*u* ranges from 0 to *k*_{0}, going through
the entire range before any new element is added to the *T*_{0}
sequence. For an *x* value in (1/2,1), the values range from 0 to
-*k*_{0}, but, again, this entire range is covered before any new
element is added to the *T*_{0} subsequence.

Let the *amplitude* of a sequence be the difference between its maximal
and its minimal value. Given that *H*_{0} goes through its entire
cycle before any element is added to *T*_{0}, the amplitude of
the total sequence is the amplitude of *H*_{0} (which we know to
be *k*_{0}) plus the amplitude of *T*_{0}. Let us
now, therefore, calculate this latter amplitude.

Let us subdivide the range (0,1) as follows:

*D*_{1}=(0,*x*]*D*_{2}=(*x*,2*x*]- ...
*D*_{k0}=(1/2-*y*-*x*,1/2-*y*]*M*=(1/2-*y*,1/2+*y*)*U*_{k0}=[1/2+*y*,1/2+*y*+*x*)- ...
*U*_{0}=[1-*x*,1)

Following the same logic, the next hit on *M* will be at an offset of
2*x* mod 2*y*, then at 3*x* mod 2*y*, etc.. In fact, if
we were to re-scale *M* into (0,1), the sequence will be
*x*_{1}, 2*x*_{1},... (mod 1), where
*x*_{1}=*x*/(2*y*) mod 1. Notably, *x*_{1}
is also an irrational.

This means that the *d*-*u* subsequence within *M* behaves
exactly as the *d*-*u* full sequence for *x*_{1}. In
particular, the two have the same amplitudes.

We can now, therefore, repeat the analysis we made with *x* on
*x*_{1}, dividing its sequence to *H*_{1}
and *T*_{1} subsequences. The amplitude of this sequence will be
*k*_{1}=1/2 div min(*x*_{1},1-*x*_{1})
plus whatever is the amplitude for the subsequence
*T*_{1}, which, in turn, can be considered as the amplitude of
the *d*-*u* values for a linear sequence based on a new irrational
value, *x*_{2},
etc.. In total, the amplitude of the original sequence is bounded from below
by an infinite sum of *k*_{i}s, each of which is a positive
integer, and so is, itself,
infinite, from which we can conclude that |*d*-*u*| is unbounded.

Q.E.D.

As can be seen, the sequence of *x*_{i}s relate to each
other by being residues in a continuous fraction expansion of *x*
(or, strictly speaking, of 2*x*).
Analysing the sequence by means of continuous fractions (which Jan Fricke
did) allows us to describe the *d*-*u* sequence very well, showing
how quickly it diverges and how many times it hits zero along the way.
These and similar follow-up questions are left for the reader.

To the best of Jan's knowledge, the problem originated from a question that was asked on the newsgroup de.sci.mathematik some years ago.