Jordan type representation for sequences with bounded variation

Jordan type representation for sequences with bounded variation

We say that a sequence $(x_n)$ of real numbers is of bounded variation iff
$ \sum_{n=1}^\infty |x_{n}-x_{n-1}|<\infty$ (where $x_0:=0$).
Let $(x_n)$ be convergent to zero.
Do there exist monotonic convergent to zero sequences $(a_n)$, $(b_n)$
such that $x_n=a_n-b_n$ ?
Thanks