- #1
Proof by induction is a mathematical method used to prove that a statement is true for all natural numbers. It involves breaking the statement down into smaller cases and proving that it holds true for the first case. Then, assuming it holds true for the nth case, it is proven to hold true for the (n+1)th case. This process is repeated until the statement is proven to hold true for all natural numbers.
The steps for using proof by induction are:
Inequalities are often used in proof by induction to show that a statement holds true for a smaller case, and then use that information to prove that it holds true for a larger case. This allows for a more efficient and concise proof. Inequalities can also be used to show that a statement is true for all natural numbers, rather than just a specific value.
Proof by induction is typically used for proving statements that involve natural numbers, sequences, and series. If a statement involves inequalities and the variable is a natural number, then proof by induction is a good method to use. Additionally, if the statement seems to have a pattern or is a generalization of a specific case, proof by induction may be a good approach.
Some common mistakes made when using proof by induction for inequalities include:
