# Methods of Proof

To prove mathematical results, in general we use any of the following methods.

**(1)** When statements of the form ** p**$$ \Leftrightarrow $$

**are used to arrive at the truth of a certain mathematical result, this kind of approach to establish the result is known as the “**

*q***direct proof**”.

**(2)** In case an equivalent statement is used to arrive at the result, then this method of proof is known as “**indirect proof**”. In particular, in place of arriving at a statement ** q** form a statement

**if we arrive at $$ \sim $$**

*p***from $$ \sim $$**

*p***then such a kind of proof is called “**

*q***contrapositive proof**”.

Symbolically, we establish $$ \sim $$** q** $$ \Rightarrow $$$$ \sim $$

**in place of**

*p***$$ \Rightarrow $$**

*p***.**

*q***(3)** Sometimes, to prove a certain result ** p**, it is convenient to prove the equivalent form $$ \sim $$($$ \sim $$

**), i.e. negation of $$ \sim $$**

*p***. This method of proof involves**

*p***contradiction of negation**of

**.**

*p*For example, to prove the uniqueness of a certain element $$x$$, if possible, suppose $$y$$ is

an element satisfying the properties of $$x$$ then we show that $$x = y$$. In such a case, instead of directly proving the proposition ** p= **“$$x$$ is unique,” i.e. for every $$y$$ satisfying properties of $$x$$, $$x = y$$, we prove $$ \sim $$($$ \sim $$

**) or $$ \sim $$($$x \ne y$$) i.e. $$x = y$$.**

*p***(4)** To prove a mathematical result ** p(n)** concerning natural number n, it is sometimes convenient to prove the truth of

**for**

*p(n)***whenever**

*n +**1***holds true for**

*p(n)***. And if**

*n***is the least natural number for which**

*m***holds, then the truth of**

*p(n)***is established for all $$n \geqslant m$$.**

*p(n)***Such a method of proof is called the “**

**method or principle of finite induction**”. This method depends upon the fact that the natural numbers follow the principle of finite induction.

To illustrate this method, let us prove for all natural numbers $$n \geqslant 3$$that

\[n > {\left( {1 + \frac{1}{n}} \right)^n}{\text{ }}\forall n \geqslant 3\,\,\,\,\, – – – \left( * \right)\]

The result (*) is evident for $$n = 3$$. Let us suppose that it is true for some $$n \geqslant 3$$. Then (*) gives that

\[n + 1 > {\left( {1 + \frac{1}{{n + 1}}} \right)^n} + 1\]

And

\[n + 1 > {\left( {1 + \frac{1}{{n + 1}}} \right)^n}\]

i.e.

\[1 > \frac{1}{{(n + 1)}}{\left( {1 + \frac{1}{{n + 1}}} \right)^n}\]

Therefore,

\[n + 1 > {\left( {1 + \frac{1}{{n + 1}}} \right)^n} + \frac{1}{{(n + 1)}}{\left( {1 + \frac{1}{{n + 1}}} \right)^n}\]

i.e.

\[n + 1 > {\left( {1 + \frac{1}{{n + 1}}} \right)^{n + 1}}\]

Thus, (*) being true for $$n \geqslant 3$$ is also true for$$n + 1$$. And it already holds for $$n = 3$$. Hence (*) is established.

The use of methods described in **(1) **to **(4) **will appear in various places in our subsequent course of study. In various types of proofs the **quantifier symbols** $$\forall $$ and $$\exists $$ are found to be quite useful. These symbols $$\forall $$ and $$\exists $$ stand for the words “**for all**” and “**there exists**,” respectively. Whenever we write$$\exists $$, it stands in the sense that we can find the quantity it precedes under specified conditions.