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Mathematical Induction Steps

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Basis Step

Show that the statement holds for the first natural number, usually 0 or 1.

Inductive Step

Assume the statement holds for some arbitrary natural number $k$ and show it holds for $k+1$ .

Inductive Hypothesis

The assumption that the statement holds for some arbitrary case $k$ .

Inductive Conclusion

Arriving at the end of the induction process, concluding the statement holds for all natural numbers.

Strong Induction, Base Case

Prove the statement for the base cases, which may include several initial natural numbers, not just the first.

Strong Induction, Inductive Step

Assume the statement holds for all natural numbers less than $k$ and show it holds for $k$ .

Induction Anchor

The choice of initial value for which the basis step establishes the truth of the statement.

Proof of Base Case

A detailed verification that the base case meets the condition of the theorem or statement being proved.

Verification of Inductive Step

Show that each case follows from the previous one, or from the collection of all previous ones in strong induction.

Conclusion of the Inductive Principle

State that, since the basis step and inductive step have been proved, by induction the statement is true for all natural numbers.

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