Weierstrass Theorem in Optimization
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.