Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 144 / 8 / i / Solution

Take $f<g$ in $I={\mathbb{Q}}^{\lambda }$ and let $\alpha$ be their first differing coordinate. Choose $r\in \mathbb{Q}$ with $f(\alpha )<r<g(\alpha )$, copy their common initial segment below $\alpha$, put $r$ at $\alpha$, and put zero at every later coordinate. The resulting eventually zero element lies strictly between $f$ and $g$, so $J$ is dense.

For each $\mu <\lambda$, the eventually zero functions supported below $\mu$ number at mostMinimality of $\lambda$ gives $2^{|\mu |}\le \kappa$ for every $\mu <\lambda$. Also $\lambda \le \kappa$, since $2^{\kappa }>\kappa$. Taking the union over $\mu <\lambda$ therefore gives $|J|\le \kappa$.