Mathematical induction

Prove a statement by checking both the first step and the link to the next.

Meaning

Mathematical induction proves a statement at a starting natural number, then shows that if it holds at an arbitrary step, it also holds at the next. This carries the result through all subsequent steps.

Example

Assume the first odd numbers sum to Adding the next odd number, gives The result also holds at so it holds for every positive integer Both the starting step and the implication to the next are needed. The induction hypothesis is a condition used to prove the next step, not an unconditional assumption that the conclusion is true.

Practice for this concept

Scope and references

Topics

Checking the initial step and proving the next step

Learning goal

Prove a statement by checking both the first step and the link to the next.

Helpful background

Review the order of natural numbers and reasoning from assumptions to conclusions.