The high-level machinery proof:

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

§ (1) Continuous image of compact set is compact

• Let $f: A \rightarrow B$ be a continuous function. Let $C \subseteq A$ be a compact set.
• Need to show $f(C)$ is compact.
• Take any open cover $\{ V_i \}$ of $f(C)$. Need finite subcover.
• Pullback $V_i$: Let $U_i \equiv f^{-1}(V_i)$. Each $U_i$ open as $f$ is continuous. $\{ U_i = f^{-1}(V_i) \}$ cover $C$ since $\{ V_i \}$ cover $f(C)$.
• Extract finite subcover $\{ U_j : j \in J \}$ for finite set $J$.
• Push forward finite subcover: $\{ V_j : j \in J \}$ cover $f(C)$ as $U_j$ cover $C$.