## Proof Exercise #3 Continuation 4:

proving the Heine-Borel Covering Theorem

Yes.

Let w be the least upper bound of M.

Case 1. w = b.

In that case, the proof is complete.

Case 2. w < b.

What, from the list below, do you think, will be our next step?

choice #1: Note that a right triangle can be isosceles.

choice #2: Note that a and b might be rational numbers.

choice #3: Note that w might be a rational number.

choice #4: Note that a < x < b.

choice #5: Note that w must be a member of some member (p,q) of G.

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Let w be the least upper bound of M.

Case 1. w = b.

In that case, the proof is complete.

Case 2. w < b.

What, from the list below, do you think, will be our next step?

choice #1: Note that a right triangle can be isosceles.

choice #2: Note that a and b might be rational numbers.

choice #3: Note that w might be a rational number.

choice #4: Note that a < x < b.

choice #5: Note that w must be a member of some member (p,q) of G.

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