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Prove or disprove:

  1. Every cauchy-sequence in $\mathbb{R}$ includes a subsequence which is monotonic.
  2. Every monotonic increasing cauchy-sequence in $\mathbb{R}$ converges to its supremum.
  1. I would say it's true because a main attribute of cauchy-sequences is that its sequences always get smaller and smaller with each other, so each one will be monotone.

  2. I say it's false but I cannot reason it :p


What do you think?

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  • $\begingroup$ I think both is true the first, because I think I recall something like every real valued sequence has a monotone subsequence ( see en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem) and the second because it converges against something and since it is monotonic increasing the limit has to be its supremum. $\endgroup$ – ctst Sep 19 '16 at 20:16
  • $\begingroup$ Your reasoning in 1. seems problematic. For one thing, the answer doesn't really depend on the sequence being Cauchy. Also, why do you think that the second is false? $\endgroup$ – John Coleman Sep 19 '16 at 20:17
  • $\begingroup$ Your reasoning for (1) makes no sense. What does "always get smaller and smaller with each other" mean? "Each what will be monotone"? There are certainly Cauchy sequences, and subsequences of Cauchy sequences, that are not monotonic. $\endgroup$ – Robert Israel Sep 19 '16 at 20:25
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1) Every Cauchy sequence converge. It's very easy to construct a subsequence that is monotonic from a sequence that converge. Let $(x_n)$ converge to $x$. Either $(\ell-\varepsilon,\ell)$ or $(\ell,\ell+\varepsilon)$ has infinitely many term of the sequence for all $\varepsilon>0$. Suppose WLOG that it is $(\ell-\varepsilon,\ell)$. Let $n_0\in\mathbb N$. By definition of the limit, there is $n_1>n_0$ s.t. $x_{n_0}\leq x_{n_1}\leq \ell$. Now, there is $n_2>n_1$ s.t. $x_{n_1}\leq x_{n_2}\leq\ell$... finally we constructed a subsequence $(x_{n_k})$ that is monotonic.

2) It's of course true. The sequence is cauchy and thus convergent. Therefore it's bounded. Let $\ell=\sup x_n$. I let you show that $(x_n)$ converge to $\ell.$

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  1. A Cauchy sequence $(x_n)$ in $\mathbb R$ converges to a limit $l$. Then at least one of the three sets $l^+=\{n \in \mathbb N \ ; \ x_n > l\}$, $l^0=\{n \in \mathbb N \ ; \ x_n = l\}$ and $l^-=\{n \in \mathbb N \ ; \ x_n > l\}$ is infinite. From there, you can find a monotonic sequence converging to $l$.

  2. Again, a Cauchy sequence converges. If it is also increasing, it converges to its supremum.

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