Show that every monotonic increasing and bounded sequence is Cauchy. The title is kind of misleading because the task actually to show

Every monotonic increasing and bounded sequence $(x_n)_{n\in\mathbb{N}}$ is Cauchy

without knowing that:


Every bounded non-empty set of real numbers has a least upper
bound. (Supremum/Completeness Axiom)   
A sequence converges if and only if it is Cauchy. (Cauchy
Criterion)
Every monotonic increasing/decreasing, bounded and real
sequence converges to the supremum/infimum of the codomain (not sure 
if this is the right word).


However, what is allowed to use listed as well:


A sequence is called covergent, if for $\forall\varepsilon&gt;0\,\,\exists N\in\mathbb{N}$ so that $|\,a_n - a\,| &lt; \varepsilon$ for $\forall n&gt;N$. (Definition of Convergence)
A sequence $(a'_k)_{k≥1}$ is called a subsequence of a sequence $(a_n)_{n≥1}$, if there is a monotonic increasing sequence $(n_k)_{k≥1}\in\mathbb{N}$ so that $a'_{k} = a_{n_{k}}$ for $\forall k≥1$. (Definition of a Subsequence)
A sequence $(a_n)_{n≥1}$ is Cauchy, if for $\forall\varepsilon&gt;0\,\,\exists N=N(\varepsilon)\in\mathbb{N}$ so that $|\,a_m - a_n\,| &lt; \varepsilon$ for $\forall m,n&gt;N$. (Definition of a Cauchy Sequence)
(Hint) The sequence $(\varepsilon\cdot\ell)_{\ell\in\mathbb{N}}$ is unbounded for $\varepsilon&gt;0$. (Archimedes Principle)


Would appreciate any help.
If $x_n$ is not Cauchy then an $\varepsilon&gt;0$ can be chosen (fixed in the rest) for which, given any arbitrarily large $N$ there are $p,q \ge n$ for which $p&lt;q$ and $x_q-x_p&gt;\varepsilon.$

Now start with $N=1$ and choose $x_{n_1},\ x_{n_2}$ for which the difference of these is at least $\varepsilon$. Next use some $N'$ beyond either index $n_1,\ n_2$ and pick $N'&lt;n_3&lt;n_4$ for which $x_{n_4}-x_{n_3}&gt;\varepsilon.$ Continue in this way to construct a subsequence.

That this subsequence diverges to $+\infty$ can be shown using the Archimedes principle, which you say can be used, since all the differences are nonnegative and there are infinitely many differences each greater than $\varepsilon$, a fixed positive number.
1)Every monotonous increasing/decreasing, bounded and real
sequence converges to the supremum/infimum of the codomain (not sure if this is the right word).

2)A sequence converges if and only if it is Cauchy. (Cauchy
Criterion).

You wrote it yourself.