Question

Prove the converse of Theorem 3.3.4 by showing that if a set

K⊂R is closed and bounded, then it is compact.

a) Bolzano-Weierstrass Theorem
b) Heine-Borel Theorem
c) Cantor's Theorem
d) Cauchy's Theorem

Answer 1

The converse of Theorem 3.3.4 is proven using the Heine-Borel Theorem (b).

Step-by-step explanation:

The correct option is b) Heine-Borel Theorem.

The Heine-Borel Theorem states that a subset of real numbers is compact if and only if it is closed and bounded.

So, to prove the converse of Theorem 3.3.4, we need to show that if a set K⊂R is closed and bounded, then it is compact.

1. Closed: Assume K is closed. This means that it contains all its limit points.
2. Bounded: Assume K is bounded. This means that there exists a constant M such that |x| ≤ M for all x∈K.
3. Now, we can invoke the Heine-Borel Theorem to conclude that K is compact.

Therefore, the converse of Theorem 3.3.4 is proven using the Heine-Borel Theorem (b).