Thread: Proof of the Least Upper Bound(Supremum) of a Set

I'm having a little bit of a problem with this; any help would be handy!

Find, with proof the least upper bound of this set of real numbers!

E = {6 - (2/(3 n + 1)) : n is a element of N}
Where N is the natural Numbers

Originally Posted by SuperFuzzyGremlin

Find, with proof the least upper bound of this set of real numbers! E = {6 - (2/(3 n + 1)) : n is a element of N}
Where N is the natural Numbers

The sequence is increasing and bounded above. So it converges. RIGHT?

Claim Sup(E) = 6

To prove this, first 6 must be found to be a valid upper bound for E. Then we must ascertain that their is no smaller upper bound then 6. Where y < 6, where y might be a smaller upper bound for E.

Proof:

Let x ∈ E with x>y
>y
>y-6
>y-6
-2>(y-6)(3n+1)
n>

Then pick
such that n>
This means that >y
Hence y can not be an upper bound for E!!!

Finally got it after pouring through a book for a very long while!!!
Thanks for the help!

Originally Posted by SuperFuzzyGremlin

Claim Sup(E) = 6

To prove this, first 6 must be found to be a valid upper bound for E. Then we must ascertain that their is no smaller upper bound then 6. Where y < 6, where y might be a smaller upper bound for E.

Proof:

Let x ∈ E with x>y
>y
>y-6
>y-6
-2>(y-6)(3n+1)
n>

Then pick
such that n>
This means that >y
Hence y can not be an upper bound for E!!!

Finally got it after pouring through a book for a very long while!!!
Thanks for the help!

this is incorrect. you assumed x > y. that is, y is not an upper bound. this is what you want to show, not assume. your choosing of n seems to be inconsequential. moreover, you ended up with exactly what you assumed, which means you've shown nothing, just went in a circle.

I would proceed this way:

Clearly 6 is an upper bound for E, since (i leave it to you to show this).

To show that , we need to show that 6 is the least upper bound. Assume, to the contrary, that it isn't. Rather, let where . Then, for all , we have .

Now, let for some . Then we have, for all , that



for all .

Which is absurd (why?). Hence our original assumption must have been in error and, in fact,