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?
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.