(2.1)From time to time, a statement A(n) will have to be proved which involves as a parameter the natural number n. For example, a statement like
(2.1)
is frequently proved by mathematical induction, that is, a proof of, if
(I) A(1) is true,
(II) it follows from A(n) that A(n+1) is true for any natural number n.
This confirms the correctness of the statement for all natural numbers. In fact, by (I), A(1) is true.If you now apply (II) with n = 1, you confirm that A(2) is true. Repeated application of (II) with increasing n establishes the truth of the statement for all natural numbers. Application of mathematical induction to (2.1) yields:
(I) For n = 1, (2.1) is true, because the number 1 is on both sides of the equation, whence A(1) is true.
(II) Assume that A(n) is true, that is
It must now be confirmed that A(n + 1), that is,
(2.2)
is true. Rewrite the left hand side of (2.2)
Since it has been
assumed already that A(n) is true, 
= n(n
+ 1)/2, whence
,
which is (2.2), so that A(n + 1) is true.
Mathematical induction may at times already start with A(0), when you confirm the validity of A(n) for all non-negative integers n.
Instead of (II), the following statement (II' )may prove more convenient:
From the truth
of A(k) for all integers k of the interval 1 
k
n
follows its correctness for A(n+1).
Obviously, (I) and (II') confirm the truth of A(n) for all natural numbers n.