# Exploring the Power of Mathematical Induction for Proving Sequences and More

• ascapoccia
In summary: Basically, the method of mathematical induction works because it can prove something is true for all n greater than or equal to a certain value. The axiom used is the least element principle, which states that given a set of integers, there must be a minimal integer, u, in the set, such that u is less than or equal to every element in the set. So, by proving that something is true for all n greater than or equal to a certain value, the method of mathematical induction proves that it is true for all n.
ascapoccia
I know this sounds kind of like a basic question, but why does the method of mathematical indcution work to prove things like sequences and such? All the other proof methods I have learned have made sense to me, and I can prove using logical truth tables or axioms, but I don't really get how this method works. Is there some special axiom I don't know about, or is there perhaps a proof that proves the method works?

My problem is not applying it, but understanding exactly how the steps I am taking prove whatever it is I am doing. I can apply it and prove things with it, but I would really like to know exactly what it is I am doing in proving a base case, assuming for n, and then proving for n+1.

Thanks in advance for answer to this question. I know it is kind of basic, but it didn't really fit the format for the Homework subforum.

The principal of mathematical induction is that for integers, if P(n) is true for the basis n=0, (or sometimes n=1 depending on how you start), and the the induction step: P(n) implies the truth of P(n+1), then the P(n) is true for all n greater than or equal to the basis.

One writer looks at it as proof by contradiction: Assume it is false for some value of n, then there must be a least value for which it is false, say n=s. This can not be the basis of the induction since we have shown that to be true. So the P(s-1) is true, but then by the induction hypothesis, P(s) is true.

The axiom used here is the Least Element Principal, which states given a collection of integers, there must be a minimal member, u, in the set, such that u is less than or equal to every element in the set.

This differs from the greatest lower bound axiom, which states every collection of, say, rational numbers, has a minimal element, u, less than or equal to every element in the set. BUT u does not have to be a member of the set. Take the set of all rational numbers greater than or equal to the square root of two, which is the greatest lower bound, but is not in the set.

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Thank you. I've been wondering about this for a while, and didn't have the heart to ask my professor what seemed should have been a fairly simple deduction.

## 1. What is Mathematical Induction?

Mathematical Induction is a method of mathematical proof used to prove statements about natural numbers or other well-ordered sets. It is based on the principle that if a statement is true for one number, and it can be shown that it is also true for the next number, then it is true for all subsequent numbers.

## 2. How does Mathematical Induction work?

The process of mathematical induction involves two steps: the base case and the inductive step. In the base case, the statement is proven to be true for the first number in the set. In the inductive step, it is shown that if the statement is true for one number, it is also true for the next number. This process is repeated until it can be concluded that the statement is true for all numbers in the set.

## 3. When is Mathematical Induction used?

Mathematical Induction is used to prove statements about natural numbers or other well-ordered sets. It is commonly used in algebra, number theory, and other branches of mathematics to prove theorems and properties.

## 4. What are the advantages of using Mathematical Induction?

Mathematical Induction is a powerful tool for proving statements about natural numbers or other well-ordered sets. It allows for the proof of general statements without having to check each individual case. It also provides a clear and logical structure for proofs.

## 5. What are some common mistakes when using Mathematical Induction?

Some common mistakes when using Mathematical Induction include assuming the statement is true without properly proving the base case, incorrectly setting up the inductive step, and using circular reasoning. It is important to carefully follow the steps of mathematical induction and to check for any errors in the proof.

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