Pioneer Continent

Why calculus must live on the real line: the rationals have holes (√2 is irrational, and {x∈Q | x²<2} has no supremum inside Q) while the least-upper-bound property of R fills them all. Master the definitions of sup/inf, the Archimedean property (for every ε>0 some n has 1/n<ε) and the density of Q in R — the foundation under every later limit argument. Classic pitfall: sup is not max — sup(0,1)=1 exists but max(0,1) does not; the infimum of a set bounded below is the greatest of its lower bounds.