Definition:Principle of Mathematical Induction

Basis for the Induction
The step that shows that the proposition is true for the first value $n_0$ is called the basis for the induction (also sometimes informally called the base case).

Induction Hypothesis
The assumption made that $P \left({k}\right)$ is true for some $k \in \N$ is the induction hypothesis (also sometimes called the inductive hypothesis).

Induction Step
The step which shows that $P \left({k}\right) \implies P \left({k+1}\right)$ is called the induction step (also sometimes called the inductive step).

Also known as
This principle is sometimes referred to as the Principle of Weak Induction.

Other sources use the term First Principle of Mathematical Induction.

Also see
Note the difference between this and the Second Principle of Mathematical Induction, which can often be used when it is not possible to prove $P \left({k+1}\right)$ directly from the truth of $P \left({k}\right)$, but when it is possible to prove $P \left({k+1}\right)$ from the assumption of the truth of $P \left({n}\right)$ for all values of $n_0$ such that $n_0 \le n \le k$.

In Equivalence of Well-Ordering Principle and Induction it is proved that this, the Second Principle of Mathematical Induction and the Well-Ordering Principle are all logically equivalent to each other.