## Proof Exercise #3 Continuation 5:

proving the Heine-Borel Covering Theorem

Yes.

However, there exists a real number y such that w < y < b and y is in (p,q).

This is just one more open interval, and so the number of open intervals

considered is still finite. Therefore y is a member of M.

Therefore:

choice #1: w is not an upper bound of M.

choice #2: b is a rational number.

choice #3: w is a rational number.

choice #4: w > b.

choice #5: w < a.

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However, there exists a real number y such that w < y < b and y is in (p,q).

This is just one more open interval, and so the number of open intervals

considered is still finite. Therefore y is a member of M.

Therefore:

choice #1: w is not an upper bound of M.

choice #2: b is a rational number.

choice #3: w is a rational number.

choice #4: w > b.

choice #5: w < a.

(end of page)

(jump to further reading option)

(return to list of proof exercises)