Sequences- Supremum

• Oct 22nd 2009, 10:28 PM
amm345
Sequences- Supremum
Is it true that for every two sequences
{xn} n goes from 1 to infinityand {yn}n goes from 1 to infinity satisfying for every n xn < yn one has

a. sup xn <or=sup yn?
b. sup xn < sup yn?

How do you prove or disprove these?
• Oct 23rd 2009, 02:49 AM
tonio
Quote:

Originally Posted by amm345
Is it true that for every two sequences

{xn} n goes from 1 to infinityand {yn}n goes from 1 to infinity satisfying for every n xn < yn one has

a. sup xn <or=sup yn?
b. sup xn < sup yn?

How do you prove or disprove these?

$\displaystyle \forall n \in \mathbb{N}\,,\,\, \frac{n-1}{n}<1\,\,,but\,\, \sup \left\{\frac{n-1}{n} \right\}_{n=1}^\infty=\sup \left\{1,1,...\right\}$

Tonio
• Oct 23rd 2009, 06:25 AM
amm345
Can you please elaborate on that?
• Oct 23rd 2009, 06:58 AM
HallsofIvy
He gave an example of sequences $\displaystyle \{x_n\}$ and $\displaystyle \{y_n\}$ with $\displaystyle x_n< y_n$ but [tex]sup \{x_n\}= sup \{y_n\}[/itex]

Perhaps a simpler example would be $\displaystyle x_n= -\frac{1}{n}$ and $\displaystyle y_n= 0$ for all n. $\displaystyle x_n< y_n$ for all n but their supremums (in this case they are both non-decreasing sequences so their supremums are just their limits- and both are 0.