Hechler forcing

A Hechler condition is a pair $(s,f)$ consisting of a finite sequence $s\in {\omega }^{<\omega }$ and a function $f\in {\omega }^{\omega }$. An extension lengthens the stem, increases the side function, and places every new stem value above the old side function. The generic union is a dominating real.

A function $d\in {\omega }^{\omega }$ dominates $h\in {\omega }^{\omega }$ when $d(n)>h(n)$ for all but finitely many $n$. A dominating real over a ground model dominates every such function belonging to that model.