Prove implication all Cauchy sequences have a limit $\to$ all monotone increasing bounded above sequences converge$

Question

So, we assume that all Cauchy sequences converge. How can we deduce ''all monotone bounded above increasing sequences converge'' from that? Any hints would be appreciated.

Answer 1

Hint If $a_n$ is monotone bounded above prove prove that it is Cauchy.

Hint 2: If you are familiar with the supremum, that give you the $N$ for each $\epsilon$.

If not prove it by contradiction.

As you said $\exists \epsilon$ such that for all $N$ there exist $m,n>N$ with $|x_n-x_m|>\epsilon$.

Construct inductively $x_{k_n}$ a subsequence so that $x_{k_n} \geq x_{k_1}+(n-1)\epsilon$.

$P(1)$ is trivial. For the inductive step, pick $N=k_n$. Then you can find $m_1> m_2> N$ such that $$ |x_{m_1}-x_{m_2}| > \epsilon $$ As the sequence is increasing, we have $$ x_{m_1} \geq x_{m_2}+ \epsilon \geq x_N+ \epsilon= x_{k_n}+\epsilon $$ So pick $k_{n+1}=m_1$.

Answer 2

Hint:

Let $M$ the least upperbound of an increasing bounded sequence $(u_n)$. For any $\varepsilon>0$, here is a term $u_{n_0}$ in the sequence that is within $\varepsilon$ of $M$. For all $n>n_0$, we have $\;u_{n_0}<u_n\le M$.