The suggestion that the thread be locked was because you were not reading the posts explaining the proofs.

Here is a detailed explanation of Fernando's proof. If there is something you do not understand,

**say exactly what is not clear**.

Let

be an arbitrary irrational number, and let

be a function which maps an element of the unit circle to an element of the unit circle.

Define the mapping by

and note that, since each element of

is uniquely determined by its angle

with the positive x axis, this mapping is the same as the mapping that Fernando has defined.

Note that we will use the notation

to abbreviate

First, we will show that if

are distinct integers (

), then

:

Now, that equals 0 or

iff

or

is irrational, but we chose them so that

and

, and so

.

Now, this means that all the elements of the sequence

are distinct, and therefore this sequence has a limit point (by sequential compactness), ie. a convergent subsequence.

Since that subsequence is cauchy, for any

there exist integers

such that

, but then note that

, so let

to have

.

__Now, note that__ is a length-preserving map (ie. it maps an interval of length

to an interval of length

).

Also, note that

maps the arc connecting the point

with

to the arc connecting

with

.

Consequently, since

and

preserves lengths, we have that

, and by the same fashion

for any

.

__Now, since all elements of __ are distinct, we have that for every

there exists an integer

, such that

, and so we have that

is dense in

, and in particular any set containing it is dense in

- and such is

.

Now, if we take

(which is clearly irrational) and take

, we get that

is dense in

, as conjectured.