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$\newcommand{\N}{\mathbf{N}}$$\newcommand{\R}{\mathbf{R}}$ Let $X=(X,\leq)$ be a Riesz space (a lattice which is also an ordered vector space over $\R$). Define $X^+=\{x\in X\colon x\ge 0\}$. Are the following statements equivalent?

  1. For all $x$ in $X^+$, $\inf \{ \frac{1}{n}\cdot x\colon n\in\N \} = 0$.
  2. If $x,y\in X^+$ and $n\cdot x\leq y$ for all $n\in \N$, then $x=0$.
  3. For all $x,y\in X^+$ with $x\neq 0$ the set $\{ \lambda\in \R\colon \lambda x \leq y \}$ has a largest element.
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At least, 1. implies 2., because $\inf \frac 1ny=0$. What makes you think these statements are equivalent? a particular example? – Davide Giraudo Jul 24 '12 at 15:22

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