After my question Question for the function $f(x)=\log\left(\frac{x^2}{x-2}\right)$, I have obtain a very good answer and I remember that I have never studied this theorem during my period at my university (1993) "extended version of Weierstrass's theorem". Obviously there are several pages on the internet that discuss the proof of the theorem. I would like to understand well the hypotheses and when and how to apply the theorem in practical cases especially for the students of an high school.
I have seen that are used the sequences and the minimum and maximum limits ($\lim \sup$, $\lim \min$) and I do not remember their use and the meaning of this concepts.
The generalization of EVT, in its simplest version, is not a particular theorem but a method which has the aim to extend the result, for continuous functions, when the hypotesis for a closed domain fails to be true.
In this particular case we are dealing with a continuous function defined on an open interval $(a,b)$ and such that
$$\lim_{x\to a^+} f(x)=\lim_{x\to b^-} f(x)=\infty$$
Under such conditions we can extend EVT to prove that the given function must have an absolute minimum. Fixed a suitable upper limit $M$, the key point is to show by IVT that a closed interval $[c,d]\subseteq(a,b)$ exists such that
$$f(c)\le M \, \land \, f(d)\le M $$
therefore $f: [c,d]\to \mathbb R$ fullfils the hypotesis for EVT and hence an absolute minimum exists also for the original function.