Weierstrass theorem

The Weierstrass theorem states that a continuous real-valued function on a compact space attains both a maximum and a minimum.

Compact space: A topological space in which every open cover has a finite subcover. Compactness is the domain condition that makes the theorem’s extrema attainable.
Extreme value theorem proof: A proof that a continuous function on a compact space attains its supremum and infimum. The proof shows how compactness converts boundedness into attained extrema.
Extreme value theorem on a closed interval: The claim that every continuous real-valued function on a closed bounded interval attains a maximum and minimum. The interval version is the standard introductory form of the theorem.
Weierstrass extreme value theorem: The real-analysis result that a continuous function on a closed bounded interval attains a maximum and minimum. This common name identifies the theorem’s classical interval formulation.
Continuous function: A function whose small changes in input produce small changes in output, expressed topologically by preimages of open sets. Continuity prevents gaps in function values that could block an extremum.
Supremum: The least upper bound of a set of real numbers, whether or not it belongs to the set. The proof starts with the supremum of the function’s image and shows it is attained.
Open interval: An interval that excludes both endpoints, such as the set of real numbers strictly between zero and one. A continuous function on an open interval can approach an extremum without attaining it.
Global optimization: The search for the best value of an objective function over an entire feasible domain. The theorem guarantees an optimizer exists when the feasible set is compact and the objective continuous.
Maximum and minimum: The greatest and least values attained by a function on a specified domain. These are the two values the theorem guarantees exist.
Bolzano–Weierstrass theorem: Every bounded sequence in Euclidean space has a convergent subsequence. A maximizing sequence yields a convergent subsequence whose limit preserves the function value.

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