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?

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- Oct 22nd 2009, 10:28 PMamm345Sequences- SupremumIs 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 AMtonio
- Oct 23rd 2009, 06:25 AMamm345
Can you please elaborate on that?

- Oct 23rd 2009, 06:58 AMHallsofIvy
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.