Prove or disprove: Two statements about Cauchy-sequences

Question

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?

Answer 1

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 $\ell.$

Answer 2

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.