# Boundedness homework help

1. Oct 31, 2011

### bugatti79

1. The problem statement, all variables and given/known data

Let $\ell_\infty \mathbb({R})$ be the set of bounded real sequences with k > 0 such that $\left | x_n \right |\le k$

a) $(n)=(1,2,3...) \notin \ell_\infty \mathbb({R})$. This is not bounded 'above'?

b) $(2n^2+1) \notin \ell_\infty \mathbb({R})$ Same answer as above?

c) $(1/n)=(1,1/2,1/3,1/4...) \in \ell_\infty \mathbb({R})$ Is bounded above?

d) $(4-1/n) \notin \ell_\infty \mathbb({R})$ Why is this not bounded? Is it because the value wll not go below 0?

2. Oct 31, 2011

### Deveno

Re: Boundedness

a) yes.

b) yes.

c) bounded above AND below: 0 < 1/n ≤ 1

d) i think there's a typo here

3. Oct 31, 2011

### Staff: Mentor

Re: Boundedness

Correct. No matter how large an M you pick, for some n, an > M.
Yes.
Yes, by 1.
Looks bounded to me. Every number in the sequence is less than 4. Why do you think it's not bounded?

4. Nov 1, 2011

### bugatti79

Re: Boundedness

Thanks guys,

Is d) bounded above AND below....because $0<(4-1/n) \le 4$...?

5. Nov 1, 2011

### Staff: Mentor

Re: Boundedness

For d, you have 3 <= 4 - 1/n < 4, with n being a positive integer.

6. Nov 1, 2011

### bugatti79

Re: Boundedness

Thanks Mark.