Proof by Induction

Induction is a technique by which the truth of a number of statements can be inferred from the truth of a few specific instances. Suppose that we want to prove a sequence of statements $P_1, P_2, P_3, \ldots$ to be true; or we want to prove a statement is true for a sequence of instances. Suppose that the following conditions holds:

1. For some $k>1$, we know that $P_1, P_2, P_3, \ldots, P_k$ are true.
2. For any $n>k$, the truths of $P_1, P_2, P_3, \ldots, P_n$ imply the truth of $P_{n+1}$.

We then use induction to show that every statement in the sequence is true. The first condition of the induction is called basis, and the second condition is called the inductive step, which is usually the major step in the proofs by induction.