Proving completeness of the real number line. I am trying to prove that the real number line $\Bbb{R}$ is complete. I know that every cauchy sequence in $R$ has a bounded monotone subsequence. Hence, if the subsequence has a limit $l$, the original sequence is also convergent to $l$. However I am having difficulty in proving two things:


*

*that there exists such an $l$. 

*that $l$ exists in $\Bbb{R}$, and not outside of it. For example, the bounded cauchy sequence $3,3.14,3.141,\dots $ does not converge in $\Bbb{Q}$


How should I go about proving this? Thanks!
 A: I will assume the Least Upper Bound property of the real numbers. I'm not sure if you are allowed to use this, but for some definitions it is built right in to the real numbers.
You've already shown that every Cauchy sequence in $\mathbb{R}$ has a bounded monotone subsequence, so let's assume wlog that the sequence $\{a_n\}_{n\in\mathbb{N}}$ is bounded increasing. We want to show that the sequence converges to $\sup\{a_n\}$. By hypothesis, let $l=\sup\{a_n\}$ which exists by the least upper bound property and the fact that $\{a_n\}$ is bounded above.
Let $\epsilon >0$ be given. There exists an $N\in\mathbb{N}$ such that $a_N>l-\epsilon$ as, if not, $l-\epsilon$ would bound $\{a_n\}$ from above which contradicts the definition of $l$. Now, $\{a_n\}$ is increasing and so for all $n>N$, we get $$|l-a_n|=l-a_n\leq l-a_N<\epsilon,$$ and so by definition of limit we have $\lim_{n\rightarrow\infty} \{a_n\}$ exists and is equal to $l=\sup_n\{a_n\}$.
A: I like to answer your second question as your first doubt should be clear by Daniel Rust's answer. $Q$ is not complete. So a Cauchy sequence in $\mathbb{Q}$ may not converge in $\mathbb{Q}$, as you have given an example. Such type of sequences will converge at some point of $\mathbb{R}$, a field extension of $\mathbb{Q}$.
A: first completeness of R
$$let\ \left \{ x_{n} \right \}\ be\ a\ cauchy\ sequence\ in\ R\\
\\
\\
i.e\ \ \ \forall \epsilon >0\ \exists N\ >0\ \ni \ \ d(x_{n}\,\ x_{m})=|x_{n}-x_{m}|<\epsilon \ \ \ ,n,m>N\\
\\
\\
so\ {\left \{ x_{n} \right \}}\ convergent\ sequence\ of\ numbers\\
\\
\\
\\
\therefore \ \ \ \therefore x_{n}\ \rightarrow x\ \ \in \ \ R\\
\\
\\
\ \ \ \ Hence\ R\ \ is\ a\ complete\ \ metric\ space$$
second  incompleteness of Q
$$let\ Q\ set\ of\ all\ rational\ number\ with\ metric\ define\ by\ \\
\\
\\
d(r_{1}\ ,\ r_{2})=|r_{1}-r_{2}|\ \ \ \ \ \ \ \ \ \ , r_{1}\ \ ,r_{2}\ \ \ \in \ Q \\$$
\
\
$$if\ \ x_{n}\ =\left [ 1+\frac{1}{n} \right ]^n\ \ \ \ \ ,\
 n=1\ ,2,..........\\
\\
\\
then\ \ \left \{ x_{n} \right \}\ \ \ is\ \ a\ cauchy\ \ sequence\
 \ \ \ and\ \ x_{n}\ \rightarrow \ e\ \ \ \ \\$$
\$$
but\ \ \ e\ \ \ \nless \ Q\ \ \ \\$$
\
$$\\Hence\ \ \ Q\ \ \ is\ \ incomplete\ \ metric\ \ \ space\\$$
