0

prove that " Prove that every sequence a_n has a monotone sub-sequence."

I tried to prove this by a proof of contradiction, so I assume there exist a sequence that doesn't have monotone sub-sequence, but I don't see how this can help me.

Diane Vanderwaif
  • 4,292
  • See here: http://math.stackexchange.com/questions/272153/illustration-proof-that-every-sequence-of-real-numbers-has-monotone-subsequence –  Sep 26 '13 at 19:25

2 Answers2

4

Proof:

We construct first a nonincreasing subsequence if possible. We call the $m$th element $x_m$ of the sequence $\{x_n\}$ a turn-back point if all later elements are less than or equal to it, in symbols if $x_m \geq x_n$ for all $n > m$.

If there is an infinite subsequence of turn-back points $x_{m_1}$, $x_{m_2}$, $x_{m_3}$, $x_{m_4}$, $\ldots$ then we have found our nonincreasing subsequence since $x_{m_1} \geq x_{m_2} \geq x_{m_3} \geq x_{m_4} \geq \ldots$

This would not be possible if there are only finitely many turn-back points. Let us suppose that $x_M$ is the last turn-back point so that any element $x_n$ for $n>M$ is not a turn-back point. Since it is not there must be an element further on in the sequence greater than it, in symbols $x_m>x_n$ for some $m>n$. Thus we can choose $x_{m_1}>x_{M+1}$ with $m_1>M+1$, then $x_{m_2}>x_{m_1}$ with $m_2>m_1$, and then $x_{m_3}>x_{m_2}$ with $m_3>m_2$, and so on to obtain an increasing subsequence $x_{M+1} < x_{m_1} < x_{m_2} < x_{m_3} < x_{m_4} < \ldots $ as required.

  • Where did you read this proof? I know it from Spivak's book, but I wouldn't know if it is his, or if he borrowed it somewhere. – Pedro Sep 26 '13 at 19:36
  • I took Real Analysis recently and we had to study this proof from Bruckner's book. –  Sep 26 '13 at 19:37
  • 1
    I see. Thanks for the info. This is a wonderful proof, elementary and powerful. – Pedro Sep 26 '13 at 19:39
  • Yes indeed. Credit goes to the authors. –  Sep 26 '13 at 19:40
  • My book author is John Petrovi, I don't if this is really his. – Diane Vanderwaif Sep 26 '13 at 22:06
  • 1
    @Diane Vanderwaif This link will help you. It is a free download of our textbook online. http://classicalrealanalysis.info/com/documents/TBB-AllChapters-Landscape.pdf –  Sep 26 '13 at 23:16
  • @Anonymous this book is great, thanks for sharing, I hvae bookmarked it, study it instead of my book from now on, thanks again – Diane Vanderwaif Sep 29 '13 at 12:19
  • @DianeVanderwaif You're welcome. My motto is "When in doubt, google it." Everything is available on the internet. –  Sep 29 '13 at 12:40
0

Let $(a_n)$ be a sequence of real numbers. Now have a look at the subsequence defined by $$b_n := \operatorname{sup}\left(a_k: k\leq n\right).$$

Andy Brandi
  • 858