# Mathematical Proof/Methods of Proof/Proof by Induction

The beauty of induction is that it allows a theorem to be proven true where an infinite number of cases exist without exploring each case individually. Induction is analogous to an infinite row of dominoes with each domino standing on its end. If you want to make all the dominoes fall, you can either:

- push on the first one, wait to see what happens, and then check each domino afterwards (which may take a long time if there's an infinite number of dominoes!)
- or you can
*prove*that if any domino falls, then it will cause the domino after it to fall. (i.e. if the first one falls then the second one will fall, and if the second one falls then the third one will fall, etc.)

Induction, essentially, is the methodology outlined in point 2.

## Parts of Induction

editInduction is composed of three parts:

- The Base Case (in the domino analogy, this shows the first domino will fall)
- The Induction Hypothesis (in the domino analogy, we assume that a particular domino will fall)
- The Inductive Step (in the domino analogy, we prove that the domino we assume will fall will cause the next domino to fall)

## Weak Induction

editWeak induction is used to show that a given property holds for all members of a countable inductive set, this usually is used for the set of natural numbers.

Weak induction for proving a statement (that depends on ) relies on two steps:

- is true for a certain base step. Usually the base case is or

- . That is, if is true, then is true.

If these two properties hold, one may induce that the property holds for all elements in the set in question. Returning to the example, if you are sure that you called your neighbor, and you knew that everyone who was called in turn called his/her neighbor, then you would be guaranteed that everyone on the block had been called (assuming you had a linear block, or that it curved around nicely).

### Examples

editThe first example of a proof by induction is always 'the sum of the first n terms:'

**Theorem 2.4.1.**For any fixed

- Proof

- Base step: , therefore the base case holds.

- Inductive step: Assume that . Consider .

So the inductive case holds. Now by induction we see that the theorem is true.

## Reverse Induction

editReverse induction is a method of using an inductive step that uses a negative in the inductive step. It is a minor variant of weak induction. The process still applies only to countable sets, generally the set of whole numbers or integers, and will frequently stop at 1 or 0, rather than working for all positive numbers.

Reverse induction works in the following case.

- The property holds for a given value, say .

- Given that the property holds for a given case, say , Show that the property holds for .

Then the property holds for all values .

Reverse induction is also usable in the general case ^{[1]}:
"to establish the validity of a sequence of propositions , it is enough to establish the following

(a) is valid for inﬁnitely many .

(b) If is valid, then so is .

It can be the case that we can easily prove and if for so for . In this case we have (a) for the infinitely many .

## Strong Induction

editIn weak induction, for the inductive step, we only required that for a given , its *immediate predecessor* satisfies the theorem (i.e., is true). In strong induction, we require that not only the immediate predecessor, but all predecessors of satisfy the theorem. The variation in the inductive step is:

- If is true for all , then is true.

The reason this is called strong induction is fairly obvious — the hypothesis in the inductive step is much stronger than the hypothesis is in the case of weak induction. Of course, for finite induction it turns out to be the same hypothesis, but in the case of transfinite sets, weak induction is not even well-defined, since some sets have elements that do not have an immediate predecessor.

## Transfinite Induction

editUsed in proving theorems involving transfinite cardinals. This technique is used in set theory to prove properties of cardinals, since there is rarely another way to go about it.

## Inductive Set

editWe first define the notion of a *well-ordered* set. A set is *well-ordered* if there is a total order on and that whenever is non-empty, there is a *least-element* in . That is, such that .

An *inductive set* is a set such that the following hold:

- (where is the least element of )
- If then such that

Of course, you look at that and say "Wait a minute. That means that !" And, of course you'd be right. That's exactly why induction works. The principle of induction is the theorem that says:

**Theorem 2.4.2.**If is a non-empty well-ordered set and is an inductive subset of then .

The proof of this theorem is left as a very simple exercise. Here we note that the set of natural numbers is clearly well-ordered with the normal order that you are familiar with, so is an inductive set. If you accept the axiom of choice, then it follows that every set can be well-ordered.

## Exercises

editFirst, we show that it holds for .

Suppose the identiy holds for some number , then

We aim to show that

it follows that

which is what we have set out to show. Since the identity holds for 1, it also holds for 2 and since it holds for 3 and so on.

First, we show that this statement holds for .

Assume that the equation holds for . Then,

We aim to show that

it follows that

which is what we set out to prove. By mathematical induction, the formula holds for all positive integers.

First, we show that this statement holds for .

Suppose it's true for . Then,

it follows that

We have shown that if it's true for then it's also true for . Now it's true for 1 (clear). Therefore it's true for all integers.

The claim is true for .

Now suppose it's true for . Then,

it follows that

We have shown that if for then it's also true for . Since it's true for 4 , it's true for all integers .

First, we show that this statement holds for .

Assume that the equation holds for . Then,

We aim to show that

it follows that

which is what we set out to prove. By mathematical induction, the formula holds for all positive integers.

First, we show that this statement holds for .

Assume that the equation holds for . Then,

We aim to show that

it follows that

which is what we set out to prove. By mathematical induction, the formula holds for all positive integers.

First, we show that this statement holds for .

Assume that the equation holds for . Then,

We aim to show that

it follows that

First, we show that this statement holds for .

Assume that the equation holds for . Then,

We aim to show that

it follows that

First, we show that this statement holds for .

Assume that the equation holds for . Then,

We aim to show that

it follows that

This problem can be solved without mathematical induction. We note that

Then, the question can be paraphrased as

which simplfies to

First, we show that this statement holds for .

Assume that the equation holds for . Then,

We aim to show that

it follows that

This can also be proven using the sum formula for a geometric series.

First, we show that this statement holds for .

Assume that the equation holds for . Then,

We aim to show that

it follows that

*Bernoulli's inequality*)

First, we show that this statement holds for .

- because

Assume that the equation holds for . Then,

We aim to show that

Multyplying by both sides, (desn't alter the sign of the inequality because , and so )

The last step relies on the fact that , since .

By mathematical induction, the formula holds for all positive integers.

First, we show that this statement holds for .

Assume that the inequality holds for . Then,

We aim to show that

We know that

Now we need to show that

it follows that

The last statement is clearly true ( ). Therefore,

Now we show that

We know that

Now we need to show that

it follows that

The last statement is clearly true ( ). Therefore,

Combining both inequalities we proved, we get the one we needed for the inductive step.

and by mathematical induction, the formula holds for all positive integers.

First, we show that this statement holds for .

- because

Assume that the inequality holds for . Then,

We aim to show that

We know that

Now we need to show that

it follows that

The last statement is clearly true. Therefore,

and by mathematical induction, the formula holds for all positive integers.

First, we show that this statement is true for . For

, and 3 is a factor of 3.

Since we have shown the base case to be true, we assume that the statement is true for where .

where (in other words, the expression is divisible by 3 for any positive integer).

Given that the base case was true for and the statement is assumed to be true for , it must also be true for which we attempt to prove now:

Therefore, the statement is true for based on the assumption that is true, and so by mathematical induction, the statement is true for all positive integers.

No Solutions Available

First we show that this statement holds for . For n=1,

- , and 4 is a factor of 4.

For n=2,

- , and 4 is a factor of 24.

Therefore, the base case holds. Now we assume that the equation above, holds for n=k, and attempt to prove that the equation holds for n=k+1.

By our inductive hypothesis above, 4 is a factor of f(k), and 4 is a factor of 4, so we know that 4 must also be a factor of . By mathematical induction, this equation holds for all positive integers.

No Solutions Available

No Solutions Available

First, we show that this statement holds for .

Assume that the inequality holds for where . Then,

- .

We have shown that if for then it's also true for . Since it's true for , it's true for all integers .