- Thread starter
- #1

#### OhMyMarkov

##### Member

- Mar 5, 2012

- 83

Let $a_n$ and $b_n$ be two sequences such that $a_n \leq b_n$ for all $n$. Let $A_n = \sup \{a_m \; | \; m \geq n\}$ and $B_n = \sup \{b_m \; | \; m \geq n\}$.

I want to prove that $A_n\leq B_n$. I attempted a proof by contradiction:

Assume $A_n > B_n$ for some $n$.

If $A_n = a_i$ and $B_n = b_j$ for some j, and $i$ not necessarily equal to $j$, then $a_i > b_j$. However, $b_i > a_i$, so that $A_n > B_n$ is not true.

But the thing is, what if the index $i$ is infinity, I'm not sure what to do there...

Any help would be appreciated!