In my whole problem, I was stuck in one direction, which is to show (i) implies (ii).
(i) Every Cauchy sequence in $\mathbb{R}$ converges to a limit in $\mathbb{R}$.
(ii) Every monotone and bounded sequence in $\mathbb{R}$ converges to a limit in $\mathbb{R}$.
Explaining in english, my idea is: Cauhcy is a bounded sequence. Given that (i) is true, i.e: this bounded cauchy sequence is convergent. Then, since not every bounded sequence is convergent, it requires some conditions so that it is convergent. One of a condition is "monotone". So, I try to use this to show (i) implies (ii). But my TA think "this is saying nothing".
If I would like to use some maths to explain, like $\epsilon$, a sequence $\{x_n\}$ etc.... How could I use it??