# Boundedness homework help

## Homework Statement

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?

## Answers and Replies

Deveno

a) yes.

b) yes.

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

d) i think there's a typo here

Mark44
Mentor

## Homework Statement

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'?
Correct. No matter how large an M you pick, for some n, an > M.
b) $(2n^2+1) \notin \ell_\infty \mathbb({R})$ Same answer as above?
Yes.
c) $(1/n)=(1,1/2,1/3,1/4...) \in \ell_\infty \mathbb({R})$ Is bounded above?
Yes, by 1.
d) $(4-1/n) \notin \ell_\infty \mathbb({R})$ Why is this not bounded? Is it because the value wll not go below 0?
Looks bounded to me. Every number in the sequence is less than 4. Why do you think it's not bounded?

Looks bounded to me. Every number in the sequence is less than 4. Why do you think it's not bounded?

Thanks guys,

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

Mark44
Mentor

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

Thanks Mark.