Finding the Supremum of S: A Proof

[Kinetica]

Homework Statement

Let S:={1-(-1)^n /n: n in N}
Find supS.

The Attempt at a Solution

Is this the right way to write the solution?
Thanks!

First, I want to show that 2 is an upper bound.
For any n in N, 1-(-1)^n /n is less or equal to 2. Thus, n=2 is an upper bound.

Second, I want to show that 2 is the least upper bound.
Assume there is an arbitrary upper bound v, such that v<2. Then, however, v< 1-(-1)^n /n for any n in N. Thus, v is not an upper bound. Hence, supS=2.

 

[micromass]

Homework Statement

Let S:={1-(-1)^n /n: n in N}
Find supS.

The Attempt at a Solution

Is this the right way to write the solution?
Thanks!

First, I want to show that 2 is an upper bound.
For any n in N, 1-(-1)^n /n is less or equal to 2. Thus, n=2 is an upper bound.

Why is 1-(-1)^n/n always less or equal than 2?

Second, I want to show that 2 is the least upper bound.
Assume there is an arbitrary upper bound v, such that v<2. Then, however, v< 1-(-1)^n /n for any n in N. Thus, v is not an upper bound. Hence, supS=2.

Why is v < 1-(-1)^n/n for any n in N?? (this is not true) You only need v<1-(-1)^n for one n in N. Which one?

 

[Gib Z]

Pretty close, hopefully the mistake was just a typo:

"Then, however, v< 1-(-1)^n /n for any n in N."

Replace "any" with "some" and it's fine. Can you see why?

EDIT: Too slow lol.