Proof by induction is a mathematical technique used to prove that a proposition is true for all natural numbers. The natural numbers are defined either as the positive integers or as the non-negative integers.
The first part of a proof by induction establishes the base case. This is the case that applies the proposition to the first natural number. The first natural number is either 0 or 1 depending on how this set of numbers is defined. In this part of the proof, the mathematician shows that the proposition is true for this number.
It would be impossible to write a new proof for each of the remaining natural numbers, because the natural numbers are infinite. And showing the proposition to be true for many of the natural numbers, such as all the numbers up to 50 or up to 100, would not be logically sufficient to prove that the proposition holds for every natural number. Thus, mathematicians perform what is called the inductive step to include all of the remaining natural numbers in the proof. This step requires the mathematician to show that if the proposition is true for a certain natural number, designated n, then it is also true for the natural number that immediately follows it, designated n+1.