# Can we create a dense set in the interval by this steps?

I have a question that is something I am wondering for some time now and I couldn't even begin answering it. I guess it could be called a riddle.

So, let $x\in[0,1]$ and $k\in(0,1)$. We begin at $x$ and we say that we make a "step up" by going to $x+k(1-x)$ and we make a "step down" by going to $x-kx$. Notice that whatever $x$ and $k$ we choose we have $x-kx\in[0,1]$ and $x+k(1-x)\in[0,1]$.

Then let $X_0=\{x\}$ and $X_{n+1}=\bigcup_{x\in X_n}\{x,x-kx,x+k(1-x)\}$. So $X_{n+1}$ is the set we get if we add the "up step" and the "down step" of every element of $X_n$ to $X_n$.

Let $X_\infty=\lim_{n\to\infty}X_n$. So my question is: Is $\overline{X_\infty}=[0,1]$?