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Proves a statement by demonstrating that if the conclusion is false, then the premise must also be false.

Directly derives the statement to be proved from axioms, definitions, and previously established theorems.

Establishes the existence of a mathematical object that satisfies a given property, without necessarily constructing the object explicitly.

Uses geometric figures and the properties of geometric objects to deduce theorems and solve problems.

Uses the behavior of functions as they tend towards infinity to establish the truth of a statement regarding growth rates or limits.

Shows that a certain condition cannot be satisfied by repeatedly reducing the problem to a smaller instance of the problem, which leads to an impossibility.

Proves that if a statement holds for a natural number $n$, and if it holds for $n$ then it holds for $n+1$, then the statement holds for all natural numbers.

Uses probability to demonstrate the existence or non-existence of a certain object or to establish the likely truth of a statement.

Assumes that the opposite of the statement to be proved is true and shows that such assumption leads to a contradiction, thereby proving the statement must be true.

A form of proof used to show that a given statement holds for all natural numbers.

A proof that uses only basic techniques of arithmetic and algebra, avoiding more advanced mathematics such as calculus or complex analysis.

Demonstrates that two sets have the same number of elements by constructing a bijective (one-to-one and onto) function between them.

Demonstrates the truth of a claim by dividing it into a finite number of cases and proving each one separately.

Shows the combinatorial equivalence between two expressions by interpreting them as counting the same set of objects in two different ways.