show the following: for all e>0, there exists N that is an element of N, natural

Question

show the following:

for all e>0, there exists N that is an element of N, natural numbers, such that for all n >= N, 1/n<e

the first N is just the element of bold N which is natural numbers. In the equation after such taht the N there is not bolded. I am also attaching a picture of this problem with the correct notation. I will award full points to whoever shows all steps.

show the following: for all e>0, there exists N that is an element of N, natural numbers, such that for all n >= N, 1/n 0, N N such that n ge N, 1/n

Explanation / Answer

1/n < e

<=> n > 1/e (since both e and n are positive)

Now by unboundedness of natural numbers given any real number x, there exists N, natural sich that N>x.

Now here x = 1/e.

Thus there exists a natural number N>1/e.

Now for all n>=N, n>=N>1/e

Hence for all n>=N n>1/e

Thus for all n>=N,1/n < e (by our observation of equality of two statements)

Thus there exists N such that for all n>=N, 1/n<e.

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