Proposition: All Cauchy sequences in R converge.
Proof: Let $a_n$ be a Cauchy sequence. Since $a_n$ is bounded, the upper and lower limits
$M = \limsup_{n→∞}(a_n)$ and $m = \liminf_{n→∞}(a_n)$ are finite. (Is this the "first" element in $a_n$? Or the greatest lower bound?)
It suffices to show that M = m. Given an $ε > 0$, there is an N∈N such that $|a_n − a_k|<ε$ whenever $n,k≥N$.
In particular, we have $|a_n−a_N|<ε$ for all $n ≥ N$. (Where is $a_N$ located?)
This means that $m_k≥a_N−ε$ and $M_k≤a_N+ε$ for $k≥N$. (How did M and m get the k from? And why?)
Consequently $M_k−m_k≤2ε$ for all $k≥N$ Hence $M−m≤2ε$ for every $ε>0$,and this is only possible if $M = m$
Question: What was the strategy used here? Informally speaking. What am I missing here?