Prove that the set of natural numbers is unbounded.

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Lesson Parent: 
Upper and Lower Bounds
Problem: 

Prove that the set of natural numbers is unbounded.

Answer: 

The set of natural numbers is not bounded above.

SAMPLE SOLUTION

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Solution: 

Look here for a description of the natural numbers and here for proofs by contradiction.

We will prove this by contradiction. We will assume that the natural numbers have an upper bound and then find a contradiction. The contradiction will be that even though we have a bounded set it has no least upper bound which violates the Axiom of Completeness.

So let's assume that the set of natural numbers has an upper bound. The Axiom of Completeness states that it has a least upper bound that we denote by $B$. This means that for every natural number $N$, <Sign in to see all the formulas>. So that also means that <Sign in to see all the formulas> and by subtracting $1$ from both sides we have <Sign in to see all the formulas>. The contradiction is that now $B-1$ is smaller than our least upper bound $B$. Therefore, $N$ cannot be bounded.