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**Spivak's proof of "A closed bounded subset of R^n is compact"**

Hi guys,

I'm currently taking a differential geometry course and decided I would read Spivak's Calculus on Manifolds, and then move on to his Differential Geometry series. There's a proof in here that feels unjustified to me, so I'm hoping you guys can point out what I'm missing. It's on p. 10 and it reads as follows:

**1-7 Corollary.**

*A closed bounded subset of ℝ*(The converse is also true (Problem 1-20).)

^{n}is compact.*If A[itex]\subset[/itex][itex]ℝ^{n}[/itex] is closed and bounded, then A[itex]\subset[/itex]B for some closed rectangle B. If [itex]\wp[/itex] is an open cover of A, then [itex]\wp[/itex] together with [itex]ℝ^{n}-A[/itex] is an open cover of B. Hence a finite number of [itex]U_1, ..., U_n[/itex] of sets in [itex]\wp[/itex], together with [itex]ℝ^{n}-A[/itex] perhaps, cover B. Then [itex]U_1, ..., U_n[/itex] cover A.*

**Proof.**The part in red is the part that I don't understand. How can we jump to saying that a finite number of open sets cover B? Isn't that sort of assuming the result?

(I ask these questions not because I doubt the veracity of Spivak's proof, but because I don't understand it.)