Simple Question About Bounded And Convergent Sequence

Question

Given $\{a_n\}$ sequence which is bounded by: $m\le a_n \le M$ and converges to $L\in\mathbb{R}$. How do I prove that $m\le L \le M$?

Thank you very much.

Brian M. Scott · Accepted Answer · 2012-04-30 17:31:38Z

up vote 4 down vote accepted

HINT: Suppose that $L>M$. Let $\epsilon=L-M$. Then there is an $n_0$ such that $L-\epsilon<a_n<L+\epsilon$ whenever $n\ge n_0$. But then ... what?

answered Apr 30 '12 at 17:31

Brian M. Scott
155k17139277

1

You're too fast... – David Mitra Apr 30 '12 at 17:32

@David: Not compared with Arturo ... – Brian M. Scott Apr 30 '12 at 17:33

I get that $M<a_n$ which is a contradiction; thank you very much . :) – Anonymous Apr 30 '12 at 17:35

copper.hat · Answer 2 · 2012-04-30 18:25:25Z

up vote 2 down vote

You could also notice that $[m,M]$ is a closed set, and $a_n \in [m,M]$, for all $n$. Hence the limit must also be in $[m,M]$.

answered Apr 30 '12 at 18:25

copper.hat
28.5k1038

2 Answers