Let S={2−(1/n) :n∈N}. Prove that sup S=2.

[amanda_ou812]

Homework Statement

Let S={2−(1/n) :n∈N}.Prove that sup S=2.

Homework Equations

I have a hard time proving sups and all the examples that I have do not ask you to prove it to an actual number

I have started something but do not know how to complete it.

The Attempt at a Solution

Let b=sup S. The s<= b for all s in S. So b is an upper bound for S. (Next, I know that I need to prove that b is the least upper bound by proving that something else cannot be the least upper bound. But, how do I get the 2 into the proof?) Thanks

 

[Sethric]

Try proof by contradiction. Assume that there exists an upper bound less than 2.

 

[kru_]

To show that b is the least upper bound, show that: given any epsilon > 0, there is an element s in the sequence S such that b - epsilon < s.

 

[HallsofIvy]

Both of those suggestions assume you have already proved that 2 is an upper bound for the sequence. That is, of course, obvious- for any n> 0, 1/n> 0, -1/n< 0 so 2- 1/n< 2.