Help understanding Proof of the equality of the normal and the extended integrals

Question

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This is a theorem from Spivak's calculus on manifolds. How is the first inquality in the last line of the proof established?

Answer 1

Spivak is summing up over all the partition of unity functions that are zero on $C$. Therefore, by the definition of a partition of unity, the sum is $1$ or less at every point of $A-C$ (less because conceivably some of the functions $\varphi$ that are nonzero on $C$ may be nonzero at points off $C$ as well).