What is Proof by Induction?

Mathematical induction is a powerful proof technique used to prove statements about all natural numbers. It works by proving a base case and then showing that if the statement holds for $n$, it must also hold for all integers bigger than $n$.

This is similar to toppling an infinite row of dominoes: push the first one, and each falls onto the next.

The structure of an inductive proof includes $3$ parts/steps:

1. The base case: a short proof that the statement is true for one or a couple initial/small values,
2. The inductive hypothesis: stating an assumption that the statement is true for some integer $k$, and
3. The inductive step: a proof that the statement also holds for $k+1$. The inductive step proof is a bit more involved than the base case proof, and also uses the assumption of the inductive step in the proof.

If both the base case and the inductive step are proven, the statement is true for all $n \in \mathbb{N}$. Induction is often used to prove formulas, properties of algorithms, and combinatorial identities.