A pseudo-rundamental sequence not equivalent to any monotone sequence

An algorithmic sequence $\varphi$ of rational numbers is called pseudo-fundamental if
$$ \forall m\rceil\rceil\exists n\forall kl(k>n\& l>n\supset|\varphi_k-\varphi_l|<2^{-m}). $$
Two sequences $\varphi$ and $\psi$ are called equivalent if
$$ \forall m\rceil\rceil\exists n\forall l(l>n\supset|\varphi_l-\psi_l|<2^{-m}). $$

A pseudo-fundamental sequence is constructed that is not equivalent to any monotonous sequence (it is the difference of two bounded increasing sequences). The construction is based on two recursively enumerable sets with incomparable degrees of unsolvability or on a weaker result proved independently.