It should be a cantor-set. Also, by construction, the length is preserved in each step, so the Haussdorff dimension should be 1.

To prove that it is in $[0,1]$, depends on how you construct your sets. If you impose the restriction that the segments in each step is within $(0,1)$, it should be trivial.

I assume you look for a general, fractal and self-similar construction.

I think you'll run into trouble by moving it half the size (it will self-intersect, I believe), but if you move it the full size, it is fine. This is easy via induction (make a picture).

EDIT: Here is a picture of my construction, each step on its own line. The picture is a bit deceptive in the end, it really should be "gray" or something.

It should be a cantor-set. Also, by construction, the length is preserved in each step, so the Haussdorff dimension should be 1.

To prove that it is in $[0,1]$, depends on how you construct your sets. If you impose the restriction that the segments in each step is within $(0,1)$, it should be trivial.

I assume you look for a general, fractal and self-similar construction.

I think you'll run into trouble by moving it half the size (it will self-intersect, I believe), but if you move it the full size, it is fine. This is easy via induction (make a picture).

EDIT: Here is a picture of my construction, each step on its own line. The picture is a bit deceptive in the end, it really should be "gray" or something.

lines[y_, n_] := 
  Table[Line[{{k/2^n, y}, {(k + 1)/2^n, y}}], {k, 0, 2^n - 1, 2}];
Graphics[ Join @@ Table[lines[-k/20, k] , {k, 1, 9}]]

It should be a cantor-set. Also, by construction, the length is preserved in each step, so the Haussdorff dimension should be 1.

To prove that it is in $[0,1]$, depends on how you construct your sets. If you impose the restriction that the segments in each step is within $(0,1)$, it should be trivial.

I assume you look for a general, fractal and self-similar construction.

I think you'll run into trouble by moving it half the size (it will self-intersect, I believe), but if you move it the full size, it is fine. This is easy via induction (make a picture).

It should be a cantor-set. Also, by construction, the length is preserved in each step, so the Haussdorff dimension should be 1.

To prove that it is in $[0,1]$, depends on how you construct your sets. If you impose the restriction that the segments in each step is within $(0,1)$, it should be trivial.

I assume you look for a general, fractal and self-similar construction.

I think you'll run into trouble by moving it half the size (it will self-intersect, I believe), but if you move it the full size, it is fine. This is easy via induction (make a picture).

EDIT: Here is a picture of my construction, each step on its own line. The picture is a bit deceptive in the end, it really should be "gray" or something.