§ Every continuous function on [a,b][a, b] attains a maximum

The high-level machinery proof:
  1. Continuous image of a compact set [a, b] under function ff is a compact set f([a, b]) (1).
  2. Compact set in R\mathbb R is closed (2) (Heine Borel).
  3. sup(f[a,b])sup(f[a, b]) is a limit point of f([a,b])f([a, b]). (4: sup is a limit point).
  4. Thus f([a,b])f([a, b]) (closed) contains sup(f([a,b])sup(f([a, b]) (limit point) (5: closed set contains all limit points).
  5. Thus ff attains maxima sup(f([a,b]))sup(f([a, b])) on [a,b][a, b].

§ (1) Continuous image of compact set is compact