The Weierstrass Theorem states that a continuous function defined on a closed and bounded interval necessarily attains a maximum value and a minimum value.
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Weierstrass Theorem
Let \( f : [a, b] \to \mathbb{R} \) be a continuous function on a closed and bounded interval \( [a,b] \subseteq \mathbb{R} \). Then \( f \) is bounded and attains its absolute maximum and minimum on \( [a,b] \).
Proof. Consider the set of values taken by \( f \) on \( [a,b] \), denoted by \( f([a,b]) \). Since \( f \) is continuous on \( [a,b] \), the image of \( f \) is closed. Moreover, since \( [a,b] \) is a closed and bounded interval, \( f([a,b]) \) is also bounded.
We define:
\[ M = \sup f([a,b]) \quad \text{and} \quad m = \inf f([a,b]). \]
Our goal is to show that there exist points \( x_M, x_m \in [a,b] \) such that:
\[ f(x_M) = M \quad \text{and} \quad f(x_m) = m. \]
Existence of the maximum
By the definition of \( M \) as the supremum, there exists a sequence \( \{ y_n \} \subseteq f([a,b]) \) such that \( y_n \to M \). This implies that there exists a sequence of points \( \{ x_n \} \subseteq [a,b] \) for which:
\[ f(x_n) = y_n \to M. \]
The sequence \( \{ x_n \} \) is contained in the compact interval \( [a,b] \); therefore, by the Bolzano–Weierstrass theorem, it admits a subsequence \( \{ x_{n_k} \} \) converging to a point \( x \in [a,b] \).
By the continuity of \( f \):
\[ f(x_{n_k}) \to f(x). \]
Since \( f(x_{n_k}) \to M \), it follows that \( f(x) = M \). Therefore, there exists at least one point \( x_M \in [a,b] \) such that \( f(x_M) = M \).
Existence of the minimum
The existence of the minimum follows by the same argument. By the definition of \( m \) as the infimum, there exists a sequence \( \{ z_n \} \subseteq f([a,b]) \) such that \( z_n \to m \), and hence a sequence of points \( \{ w_n \} \subseteq [a,b] \) for which:
\[ f(w_n) = z_n \to m. \]
Applying the Bolzano–Weierstrass theorem again, there exists a subsequence \( \{ w_{n_k} \} \) converging to a point \( x' \in [a,b] \). By continuity:
\[ f(w_{n_k}) \to f(x'). \]
Since \( f(w_{n_k}) \to m \), it follows that \( f(x') = m \), so there exists a point \( x_m \in [a,b] \) such that \( f(x_m) = m \).
We have thus proved that a continuous function \( f \) on a closed and bounded interval \( [a,b] \) is bounded and attains its maximum and minimum values at least at one point in \( [a,b] \).