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Just a very quick argument which reduces the possibilities: Let $\Omega\subset{\mathbb R}\cup\{\pm\infty\}$ be the $\omega$-limit set of the sequence, that is the set of limits of "converging" sub-sequences. It is a non-void closed set by construction. The property $x_{a+1}-x_a\rightarrow0$ tells us that $\Omega$ is a connected set. The property $x_{2a}-x_a\rightarrow0$ tells us that $2\Omega=\Omega$. Therefore $\Omega$ can only be equal to one of the four sets $$\{0\},\quad[0,+\infty],\quad[-\infty,0],\quad{\mathbb R}.$$