The Weierstrass Theorem states that a continuous function defined on a closed and bounded interval necessarily attains a maximum value and a minimum value.
Table of Contents
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 the function \( f \) on \( [a,b] \), which we denote 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 a bounded set.
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 of values \( \{ y_n \} \subseteq f([a,b]) \) 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 \( f \), we have: \[ f(x_{n_k}) \to f(x). \] But 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
Now we prove the existence of the minimum using the same procedure. By the definition of \( m \) as the infimum, there exists a sequence \( \{ z_n \} \subseteq f([a,b]) \) such that \( z_n \to m \). Therefore there exists a sequence of points \( \{ w_n \} \subseteq [a,b] \) for which: \[ f(w_n) = z_n \to m. \] In this case as well, the sequence \( \{ w_n \} \) is contained in \( [a,b] \). Applying the Bolzano-Weierstrass theorem again, there exists a subsequence \( \{ w_{n_k} \} \) that converges to a point \( x' \in [a,b] \).
By the continuity of \( f \), we have: \[ f(w_{n_k}) \to f(x'). \] Since \( f(w_{n_k}) \to m \), it follows that: \[ f(x') = m. \] Consequently, there exists a point \( x_m \in [a,b] \) such that \( f(x_m) = m \).
We have proven that the continuous function \( f \) defined on a closed and bounded interval \( [a,b] \) is bounded and attains its maximum and minimum values at least at one point in \( [a,b] \).