Let $(G,+,\le)$ be a partially ordered group (with identity $0$) and suppose that for each positive $g \in G$ there exists $g^\prime \in G$ such that $0<g^\prime<g$.
Is it true that there exists a sequence $(g_n)_{n\ge 1}$ of positive elements of $G$ such that $\inf \{g_n\}=0$?
Let $G\subseteq{^{\omega_1}\Bbb Z}$ be the set of $\omega_1$-sequences $\langle x_\xi:\xi<\omega_1\rangle$ of integers such that $\{\xi<\omega_1:x_\xi\ne 0\}$ is finite; the operation is coordinatewise addition. $G$ is linearly ordered by the lexicographic order.
Let $\left\langle x^{(n)}:n\in\Bbb N\right\rangle$ be a sequence of positive elements of $G$. For $n\in\Bbb N$ let $$S_n=\left\{\xi<\omega_1:x_\xi^{(n)}\ne 0\right\}\;,$$
and let $S=\bigcup_{n\in\Bbb N}S_n$. Then $S$ is countable, so there is $\alpha<\omega_1$ that is greater than any element of $S$. Define $x\in G$ by
$$x_\xi=\begin{cases} 1,&\text{if }\xi=\alpha\\ 0,&\text{otherwise}\;; \end{cases}$$
then $0_G<x<x^{(n)}$ for each $n\in\Bbb N$, so $\inf\limits_{n\in\Bbb N}x^{(n)}>0_G$.