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Can anyone help me verify a piece of a proof which shows that normality implies that any point-finite open covering can be shrunk?

To be more specific, the proof is in Willard's General Topology: link to Google books. I cannot verify that the $F_{\alpha}$ which they define inductively is contained in the $U_{\alpha}$. In fact, by drawing a picture, I think I can find a counterexample. However, I think Willard knows what he's doing so there's likely something I don't see.

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