I'm studying for my analysis midterm and ran across the following in my notes:
For a sequence $\{x_n\}$ in $\mathbb{R}$, define $b_k = \sup \{x_{k+1}, x_{k+2}, \ldots\}$ and $a_k = \inf \{x_{k+1}, x_{k+2}, \ldots \}$. We have $b_k \le b_{k+1}$ and $a_k \ge a_{k+1}$.
I think I must have made a mistake when copying this from the board because it seems to me that it should be $b_k \ge b_{k+1}$ and $a_k \le a_{k+1}$ as we are throwing away potential upper-bounds and lower-bounds, respectively.