Nets and Real Numbers

[SlipEternal]

Let \(\displaystyle \psi:[0,\infty)\to \mathbb{R}\) be a net such that for all \(\displaystyle \epsilon>0\), there exists \(\displaystyle \alpha \in [0,\infty)\) such that for all \(\displaystyle \beta,\gamma \in [\alpha,\infty)\), \(\displaystyle |\psi(\beta)-\psi(\gamma)|<\epsilon\). Does that imply that all subsequences are Cauchy? Or does there exist such a net that does not converge in the reals?

 

[SlipEternal]

Slightly related: Is \(\displaystyle \mathbb{Q}\times [0,1)\) with the dictionary order and the induced order topology a metrizable space? I know that \(\displaystyle \mathbb{N}\times[0,1)\) with the same order and induced topology is isometric to the reals, so it would stand to reason that the former would be metrizable.