Lee's Proof on Top Cohomology of Orientable Noncompact, Connected Manifold

Question

My question is about the theorem $17.32, pp. 455-456$ in Lee's book "Introduction to Smooth Manifolds," second edition. With the hypotheses as in the title of the question, the conclusion is that $H_{dR}^n(M)=0,$ where $M$ is a smooth manifold of dimension $n.$ The proof begins with a choice of exhaustion function $f\in C^\infty (M)$ which can be taken to map $M$ onto $[0,\infty).$ For each integer $i>0,$ one defines open sets $V_i=f^{-1}((i-2,i)).$ These cover $M$. The argument that follows relies on the previously proved theorem $17.30$, $pp. 454-455,$ which would seem then to require that the $V_i$ be connected. But I do not see why the $V_i$ as defined here, must be so. I am probably missing something obvious, but I can't patch up the proof without applying $17.30$ and I cannot adjust the $f_i$ so that the $V_i$ are necessarily connected. I am thinking that perhaps this is one of those proofs that relies on a technical topological lemma, and that as Lee does not cite it, it is probably a triviality. Can someone point me in the right direction?

Answer 1

The proof is wrong. See Jack's errata, pages 7-8.