  This note tries to show that a re-examination of a first course in analysis,
using the more sophisticated tools and approaches obtained in later stages, can
be a real fun for experts, advanced students, etc. We start by going to the
extreme, namely we present two proofs of the Extreme Value Theorem: "the
programmer proof" that suggests a method (which is practical in down-to-earth
settings) to approximate, to any required precision, the extreme values of the
given function in a metric space setting, and an abstract space proof ("the
level-set proof") for semicontinuous functions defined on compact topological
spaces. Next, in the intermediate part, we consider the Intermediate Value
Theorem, generalize it to a wide class of discontinuous functions, and
re-examine the meaning of the intermediate value property. The trek reaches the
final frontier when we discuss the Uniform Continuity Theorem, generalize it,
re-examine the meaning of uniform continuity, and find the optimal delta of the
given epsilon. Have fun!
