Questions on inductive definitions in a proof

In summary, the conversation discusses a problem from Kenneth Ross's book Elementary Analysis and its solution using induction. The author uses a statement P(n) to construct the terms of a non-decreasing sequence in a bounded nonempty subset of \mathbb{R} with \mbox{sup }S\notin S, ultimately proving that there is a sequence (s_n) where \lim s_n =\mbox{sup }S. The author's proof involves showing that the first term exists and then using induction to show that all terms exist.
  • #1
issacnewton
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29
Hi

I was trying to solve the following problem from Kenneth Ross's Elementary Analysis book.
here is the problem.

Let S be a bounded nonempty subset of [itex]\mathbb{R}[/itex] and suppose that
[itex]\mbox{sup }S\notin S[/itex]. Prove that there is a non decreasing sequence
[itex](s_n)[/itex] of points in S such that [itex]\lim s_n =\mbox{sup }S [/itex].

Now the author has provided the solution at back of the book. I have attached the snapshot of the proof. I am trying to understand it. He is using induction here in the proof. Now in induction, we usually have a statement P(n) , which depends upon the natutal number n. And then we use either weak or strong induction. So what would be P(n) in his proof. I am trying to understand the logical structure of the proof. Thats why I decided to post in this part of PF.

thanks
 

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  • #2
It's induction in the sense that given the n-1 term he can construct the nth term. He starts with the 1st term and shows you how to construct the 2nd term. He could then, just as well, have said "proceeding in this manner". Notice that below he just says "therefore the construction continues".
 
  • #3
Sorry, maybe I didn't answer your question. He shows that the first term exists. Then he shows that given that the n-1 term exists then the nth term exists by his construction. Therefore all terms exist.
 

1. What is an inductive definition in a proof?

An inductive definition in a proof is a way of defining a concept or object by breaking it down into simpler, more basic elements and building it up from there. It is used to prove statements about a specific type of object or concept by showing that it holds true for the simplest form of that object and then using that as a basis for proving it for more complex forms.

2. How is an inductive definition different from a deductive proof?

An inductive definition is different from a deductive proof in that it starts from specific, concrete examples and uses them to prove a general statement, while a deductive proof starts from general statements and uses them to prove specific examples. Inductive definitions are often used in mathematics and computer science, while deductive proofs are more commonly used in logic and philosophy.

3. What are some common examples of inductive definitions in a proof?

Some common examples of inductive definitions in a proof include the definition of natural numbers, where the first number is given and each subsequent number is defined as the previous number plus one; the definition of a tree in graph theory, where a tree is a connected graph with no cycles; and the definition of a proof by mathematical induction, where the proof starts with a base case and then uses that to prove the statement for all other cases.

4. How do you know when to use an inductive definition in a proof?

An inductive definition in a proof is typically used when trying to prove a statement about a specific type of object or concept that can be broken down into simpler components. It is also useful when trying to prove a statement for all cases of that object or concept, rather than just a specific example. Additionally, inductive definitions are commonly used in situations where the concept being defined is recursive in nature.

5. What are some potential challenges when using inductive definitions in a proof?

Some potential challenges when using inductive definitions in a proof include ensuring that the base case is valid, as well as making sure that the inductive step is logically sound. It can also be difficult to determine the appropriate level of generality for the inductive definition, as defining the concept too narrowly may not cover all cases, while defining it too broadly may make the proof more complex. Additionally, it can be challenging to identify and prove the underlying properties or assumptions that are necessary for the inductive definition to hold.

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