§ Every continuous function on attains a maximum
The high-level machinery proof:
- Continuous image of a compact set
[a, b] under function is a compact set
f([a, b]) (1).
- Compact set in is closed (2) (Heine Borel).
- is a limit point of . (4:
sup is a limit point).
- Thus (closed) contains (limit point) (5: closed set contains all limit points).
- Thus attains maxima on .
§ (1) Continuous image of compact set is compact
- Let be a continuous function. Let be a compact set.
- Need to show is compact.
- Take any open cover of . Need finite subcover.
- Pullback : Let . Each open as is continuous. cover since cover .
- Extract finite subcover for finite set .
- Push forward finite subcover: cover as cover .