Maybe this doubt is silly, but I do not understand the final step of the proof of Lemma 5.2 in Hamkins' paper *Fragile measurability,* Journal of Symbolic Logic 59 (1994) 262-282.

There, $\mathbb P_\lambda$ is the $\lambda$-th step of an iterated forcing construction which is the inverse limit of the previous steps, and $G$ is a $V$-generic filter. More specifically, $\mathbb P_\lambda$ is the canonical Easton support iteration which forces the GCH below $\lambda$, and $\lambda$ is a strong limit cardinal of countable cofinality.

We have a condition $r\in\mathbb P_\lambda$ (in the proof it is called $j(p)$) such that for all $\beta<\lambda$, $r(\beta)\in G(\beta)$. My doubt is why this implies that $r\in G$.

I can prove that $r|_\beta\in G_\beta=i_{\beta\lambda}^{-1}(G)$ for all $\beta<\lambda$, and maybe the conclusion is trivial, but I do not see it. In fact, I know very few about generic filters on inverse limits and their relation with previous steps.