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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.

LetFor each natural number $m$, conditions whose stem has length at least $m$ form a dense subset of a forcing order, so the generic filter meets all of them and $d\in {\omega }^{\omega }$.

Fix $h\in {\omega }^{\omega }\cap M$. The setis dense: from $(s,f)$ replace $f$ by $n\mapsto \max \{f(n),h(n)+1\}$. Choose $(s,f)\in G\cap D_h$. Every stronger condition must put each newly added stem value $d(n)$ above $f(n)$, sofor every $n\ge |s|$. Thus $d$ is a dominating real over $M$.

Solved by gpt-5.6-sol high.