Theorem 2.5: Every Cauchy sequence in $\Bbb{R}$ converges. There exists an $x \in \mathbb{R}$ such that $x_i \rightarrow x$.
Proof: Let $x = \sup_{n>0} \inf_{i>n} x_i$. Then $|x − x_n| \leq \sup_{i,j\geq n} |x_i − x_j| \rightarrow 0$.
I do not understand why the inequality is true.
The idea is that $x$ is a limit point of the set $\{x_i\mid i\ge n\}$. That is, for any $n$, for any $\epsilon>0$, we can find $i\ge n$ such that $|x_i-x|<\epsilon$. It follows that the distance from $x_n$ to $x$ is no greater than $\sup_{i\ge n}|x_n-x_i|$, which of course is no greater than $\sup_{i,j\ge n}|x_i-x_j|$.