If f(x) is continuous in [a,b] there exists a point c in the closed interval [a,b] such that for all x in [a,b], f(x) cannot be greater than f(c). Also, there exists a point d in the interval [a,b] such that for all x in [a,b], f(x) cannot be less than f(d).

One of the more complicated theorems I have come across, the Extreme Value Theorem (single-variable, real functions only) requires a proof that I don't see in my textbook.

I've searched for it everywhere, and the only proof I have found is on wikipedia but the proof is to complicated for me to understand (I'm in 11th grade but I have been able to understand the other theorems' proofs no idea why this one should be such a stumper).

Could someone sort of breakdown the reasoning of the proof in simpler terms in case that's possible?