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

  • $\begingroup$ 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, 2018 at 19:03
  • $\begingroup$ I don't see what you consider circular here. $\endgroup$ Aug 23, 2018 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 $\endgroup$ Aug 23, 2018 at 8:15

1 Answer 1


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.


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