Understanding Boundedness in Real Sequences

[bugatti79]

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?

 

[Deveno]

a) yes.

b) yes.

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

d) i think there's a typo here

 

[Mark44]

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, a_n > 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?

 

[bugatti79]

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]

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

 

[bugatti79]

Thanks Mark.