Prove that if there is a largest natural number then it is u

Prove that if there is a largest natural number, then it is unique. Prove that if there is a largest natural number, then it is 5. Using parts (a) and (b) and Theorem 2.1.18, prove that there is no largest natural number.

Solution

if you read all three parts then you will find 5 is not exactly the largest natural number

in part (a) you have to show that largest natural number is unique.

you asked for (b) so i think you already proved for part (a)

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So on the idea as discussed above, 5 is largest natural numbers for set of numbers x<=5

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part(c)

we know if n is natural number then (n+1) is also a natural number

plug n=5

we get n=+1=5+1=6

which is clearly greater than 5

that means we can always find a largest natural number for given set.

Hence there doesn\'t exist any largest natural number.