f: (from wikipedia -- math proof methodology)

"Direct proofMain article: Direct proof
In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.[11] For example, direct proof can be used to establish that the sum of two even integers is always even:

Consider two even integers x and y. Since they are even, they can be written as x=2a and y=2b respectively for integers a and b. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clear x+y has 2 as a factor and therefore is even, so the sum of any two even integers is even.
This proof uses definition of even integers, as well as distribution law.

Proof by mathematical inductionMain article: Mathematical induction
In proof by mathematical induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often infinite) series of other cases.[12] Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number. A subset of induction is infinite descent. Infinite descent can be used to prove the irrationality of the square root of two.

The principle of mathematical induction states that: Let N = { 1, 2, 3, 4, ... } be the set of natural numbers and P(n) be a mathematical statement involving the natural number n belonging to N such that

(i) P(1) is true, i.e., P(n) is true for n = 1
(ii) P(n + 1) is true whenever P(n) is true, i.e., P(n) is true implies that P(n + 1) is true.
Then P(n) is true for all natural numbers n.

Mathematicians often use the term "proof by induction" as shorthand for a proof by mathematical induction.[13] However, the term "proof by induction" may also be used in logic to mean an argument that uses inductive reasoning."

"We have seen the enemy, and he is us!" (POGO)