Weierstrass Theorem in Optimization 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?
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.