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?