# Proof about a function from the naturals.

• cragar
In summary: If f(x+1)>=a, then it is larger than a and so it is mapped to a. If f(x+1)<=a, then it is smaller than a and so it is mapped to f(x).
cragar

## Homework Statement

Prove that there is no function f:N→N such that
for all $n \in N$ , f(n)>f(n+1).
There is no infinite decreasing sequence of naturals.

## The Attempt at a Solution

Let's assume that their is a function f(n)>f(n+1) for all n.
This would also imply that f(n) will be mapped to all of N using all n in N.
There exists some x in N such that f(x) equals the first natural number.
if f(x) gets mapped to the first natural then f(x+1) gets mapped to some natural.
This is either the first natural or some larger natural. But this is a contradiction
because we assumed that f(n)>f(n+1). so therefore no function exists.

cragar said:
There exists some x in N such that f(x) equals the first natural number.

Why?

We are assuming that their is a function f that maps
N to N. so their must exist an x in N so that f(x) equals the first natural number.
I guess this is something we are assuming.

I don't see in the problem statement that f is supposed to be surjecive. So I don't think you can assume that anything maps to the first natural number.

No f:N→N does not mean the function is "onto"
just suppose
f(1)=a
there are a-1 smaller natural numbers so one of the a numbers
f(2),f(3),f(4),...,f(a-2),f(a-1),f(a),f(a+1)
must not be
not to mention all higher terms

You are onto something though, there does exist a minimum m so that
f(n)>=m
so f cannot continue decreasing

ok thanks for the replies. So I just assume their exists an x such that
f(x)=a where a is the smallest element. then f(x+1) either gets mapped to
a or something larger than a which is a cont-radiation.

Yes. You might want to specifically mention the "well-ordering principle": every subset of N has a smallest member. Let a be the smallest member of the image of N under function f. There exist some x in N such that f(x)= a. Then look at f(x+1).

## 1. What is a function from the naturals?

A function from the naturals is a mathematical relationship between two sets of natural numbers, where each input from the first set (called the domain) is mapped to a unique output from the second set (called the range). In other words, for every input, there is only one corresponding output.

## 2. How can you prove a function from the naturals?

To prove a function from the naturals, you need to show that for every input in the domain, there is a unique output in the range. This can be done by providing a clear and logical explanation or by using mathematical notation and equations.

## 3. What is an example of a function from the naturals?

An example of a function from the naturals is the function f(x) = x + 2, where the domain is all natural numbers and the range is all natural numbers greater than or equal to 2. For example, if we input 3, the output would be 5 (3 + 2 = 5).

## 4. Can a function from the naturals have more than one output for a given input?

No, a function from the naturals can only have one output for a given input. This is a fundamental property of functions and is known as the "vertical line test". If a vertical line crosses the graph of a function more than once, it is not a function.

## 5. How are functions from the naturals used in real life?

Functions from the naturals are used in various fields such as science, engineering, and economics. They can be used to model and solve real-world problems, make predictions, and analyze data. For example, a function from the naturals can be used to calculate the growth rate of a population over time or to determine the optimal production level for a company.

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