2
$\begingroup$

Weierstrass Theorem states that any bounded sequence has a convergent subsequence.

I did that in my maths course and understood it completely. But when I was learning optimization techniques in economics, the definition by the book Sundaram was tweaked slightly to make it fit into the understanding of optimization. It goes like this - (see Figure 1)

If you read it, it is saying just what original theorem said. The only difference is that, instead of starting off with a bounded sequence and reaching the conclusion that it will have convergent subsequence, it started off with a convergent subsequence (by assuming Compact set) and reached the conclusion that it should be bounded. It makes sense right? It did to me too until I found a result on compact set, stated on the same book. (See figure 2)

This redefining of the theorem made it a circular definition - if a set can only be compact if n only if it is bounded then by assuming Compactness in redefinition we implicitly assumed boundedness.

So, my question is please tell me where am I going wrong. Have I understood the redefinition incorrectly or missed so point?

Figure 1 Figure 2

$\endgroup$
  • $\begingroup$ As a matter of writing style, may I suggest that you should perhaps avoid the excessive use of the pronoun "it", whose referent is extremely unclear, especially in the second and third paragraphs. $\endgroup$ – Herr K. Aug 22 '18 at 19:03
  • $\begingroup$ I don't see what you consider circular here. $\endgroup$ – Michael Greinecker Aug 23 '18 at 4:04
  • $\begingroup$ Compactness needs boundedness and closed bounded sequence is compact. If the 'let us consider the compact set' took care of the closed and bounded set (already a theorem [1.21] which established their relationship), then concluding it with the fact that the set will have a maximum and minimum is obvious. This redefinition took away the energy of the original Weierstrass Theorem. @Michael Greinecker $\endgroup$ – Elina Gilbert Aug 23 '18 at 8:15
4
$\begingroup$

What the book calls the "Weierstrass Theorem" is more commonly known as the Extreme Value Theorem, which states that a continuous function defined on a compact domain attains a maximum and a minimum.

A common proof of this theorem involves the use of the Bolzano–Weierstrass theorem, which you learned in your math course, and which says every bounded sequence has a convergent subsequence.

So the two theorems are actually different, albeit related, statements.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.