Let $(E,\left \| . \right \|,\sqsubseteq )$ be a partially ordered Banach space. Let $(x_n)_{n\in\mathbb N}$ a monotone sequence (let's say $u_n\leq u_{n+1}$), such that ${u_n}$ has a convergent subsequence.
I'm wondering, in which conditions the whole sequence $u_n$ is convergent?
Edit: Here, $E$ is ordered with the order relation $\sqsubseteq $ induced by a cone $ K=\{x\in E: x\geq 0\}$.