An element of $\ell^2$ wanted

Question

I am looking for an element $x=(x_0,x_1,x_2,\cdots)$ in $\ell^2$ such that the sequence $z_n, n=0,1,2,\cdots$ defined by $$z_n=2^n(x_n, x_{n+1},\cdots)$$ is dense in $\ell^2$. It seems that this is hard to do, but such $x$ exists by an indirect method (using Birkhorff transitivity theorem), see Dynamics of Linear Operators by Bayart and Matheron, p. 6.