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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.

Kevin Sheng
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  • Right, $b_k$ is non-increasing, and $a_k$ non-decreasing. – Daniel Fischer Oct 18 '14 at 23:19
  • So I made a transcription error and it should be $b_k \ge b_{k+1}$ and $a_k \le a_{k+1}$? – Kevin Sheng Oct 18 '14 at 23:26
  • It should be $b_k \geqslant b_{k+1},, a_k \leqslant a_{k+1}$, and probably you made a transcription error. It is however possible that the lecturer made a mistake when writing it on the board, so I can't authoritatively speak about the cause. – Daniel Fischer Oct 18 '14 at 23:29

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