## Proof Exercise #3 Continuation 3:

proving the Heine-Borel Covering Theorem

Yes.

Let us call the set of such x M.

We have already established that M is non-empty.

Case 1. M has no upper bound.

In that case, b is a member of M and the proof is complete.

Case 2. M has an upper bound.

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

choice #1: Take the squares of a and b and apply the Distributive Proerty.

choice #2: Draw a parallelogram, suitably positioned.

choice #3: Estimate the value of ab.

choice #4: Apply the Pythagorean Theorem.

choice #5: Apply the Least-Upper-Bound Axiom.

(end of page)

(jump to further reading option)

(return to list of proof exercises)

Let us call the set of such x M.

We have already established that M is non-empty.

Case 1. M has no upper bound.

In that case, b is a member of M and the proof is complete.

Case 2. M has an upper bound.

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

choice #1: Take the squares of a and b and apply the Distributive Proerty.

choice #2: Draw a parallelogram, suitably positioned.

choice #3: Estimate the value of ab.

choice #4: Apply the Pythagorean Theorem.

choice #5: Apply the Least-Upper-Bound Axiom.

(end of page)

(jump to further reading option)

(return to list of proof exercises)