# Prove that the natural numbers are unbounded

Natural numbers, which include all positive integers from 1 to infinity, are a component of the number system. Natural numbers, which do not include zero or negative numbers, are also known as counting numbers. They are only positive integers, not zero, fractions, decimals, or negative numbers, and they are a component of real numbers.

Numbers may be found everywhere around us, used for counting items, representing or trading money, gauging temperature, telling time, and so on. These numbers are referred to as **natural numbers** since they are used to count items. Also, a collection of all whole numbers except 0 is referred to as natural numbers. These figures play an important role in our daily actions and communication.

**What are Natural Numbers?**

Natural numbers are those that can be counted and are a component of real numbers. Only positive integers, such as 1, 2, 3, 4, 5, 6, etc., are included in the set of natural numbers. Non-negative integers are also known as natural numbers (all positive integers).

23, 56, 78, 999, 100202, and so forth are a few examples.

**Set of Natural Number**

A collection of elements is referred to as a set. In mathematics, the set of natural numbers is expressed as {1,2,3,…}. The set of natural numbers is represented by the symbol N.

**N = {1,2,3,4,5,…∞}**

**Smallest Natural Number**

1 is the smallest natural number. We know that the smallest element in N is 1 and that we can talk about the next element in terms of 1 and N for any element in N (which is 1 more than that element). Two is one greater than one, three is one greater than two, and so on.

### Properties of Natural Numbers

The four operations on natural numbers, addition, subtraction, multiplication, and division, resulting in four main characteristics of natural numbers, which are illustrated below:

**Closure Property: **A natural number is always the sum and product of two natural numbers. When it comes to addition and multiplication, the set of natural numbers, N, is closed, but not when it comes to subtraction and division.

**Associative Property: **Even if the order of the numbers is modified, the sum or product of any three natural numbers stays the same. The associative property of N says that a+(b+c) = (a+b)+c and a×(b×c) = (a×b)×c for any a, b, c ∈ N. When it comes to addition and multiplication, the set of natural numbers N is associative, but not when it comes to subtraction and division.

**Commutative Property: **Even if the sequence of the numbers is changed, the sum or product of two natural numbers stays the same. The commutative property of N says that a+b = b+a and a×b = b×a for any a, b ∈ N. When it comes to addition and multiplication, the set of natural numbers N is commutative, but not when it comes to subtraction and division.

**Distributive Property: **Natural number multiplication is always distributive over addition. For instance, a × (b + c) = ab + ac. Natural number multiplication is also distributive over subtraction. For instance, a × (b−c) = ab−ac.

### Prove that the natural numbers are unbounded

**Proof:**

The Supremum Property states that every nonempty set of real numbers bounded above contains a supremum, which is a real number. There is an infimum, which is a real number, in any nonempty set of real numbers that is bounded below. The supremum property can be used to prove other real-number characteristics.

Assume N is constrained above. Then, according to the supremum characteristic, the lowest upper bound “s” exists for every n ∈ N. Consider “k” to be the largest natural integer that is less than s. Then k + 1 > s, and s is not an upper bound of N because k + 1 is a natural integer. As a result of this contradiction, we may deduce that N has no upper bound.

### Similar Questions

**Question 1: What is the difference between whole and natural numbers?**

**Solution:**

Only positive integers are included in natural numbers, which range from 1 to infinity. Whole numbers, on the other hand, are a mix of zero and natural numbers that begin at 0 and terminate at an infinite value.

**Question 2: Is ‘0’ a Natural Number?**

**Solution:**

‘No,’ is the response to this question. Natural numbers, as we already know, range from 1 to infinity and are positive integers. However, when we combine 0 with a positive integer such as 10, 20, or any other number, we get a natural number. In reality, 0 is a full number with no meaning.

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