In Claim 5.5 of the book $ Almost$ $Free$ $Modules$, second edition, by P. Eklof and A. Mekler (page 254), the authors consider regular cardinals $\kappa$ and $\aleph_\alpha$, where $\kappa<\aleph_\alpha$.
Let $\langle\lambda_i: i<\kappa\rangle$ be a strictly increasing sequence of regular cardinals (each greater than $\aleph_\alpha$) cofinal in $\aleph_{\alpha+\kappa}$.
In the proof, the authors assert the existence of a bijection $\Psi:\kappa\times\aleph_{\alpha+\kappa}\to\aleph_{\alpha+\kappa}$ such that for $i<j<\kappa$ and $\nu\in\lambda_i$ and $\nu'\in\lambda_j$, we have $\Psi(i,\nu)<\Psi(j,\nu')$.
What is such a bijection?
Split $\aleph_{\alpha+\kappa}$ into intervals of size $\kappa$, and consider them in the lexicographic order.
In other words, look at the ordinal $\omega_{\alpha+\kappa}\cdot\kappa$, it has cardinality $\aleph_{\alpha+\kappa}$, pick a bijection witnessing that and use it at your leisure.