http://en.wikipedia.org/wiki/Least-upper-bound_property#Proof_using_Cauchy_sequences
This proof of the Least Upper Bound Property defines sequences $A_1, A_2,... $ and $B_1, B_2,...$ recursively: Let S be a nonempty set of the reals that is bounded above. Let $B_1$ be an upper bound of S, and let $A_1$ be an element of S that is not an upper bound.
1)Compute $(A_n +B_n)/2$.
2) If it is an upper bound of S, then take $A_{n+1}=A_n$ and $B_{n+1}=(A_n +B_n)/2$.
3) Otherwise, there is an $s\in S$ such that $(A_n +B_n)/2. Take $A_{n+1}=s$ and $B_{n+1}=B_n$.
How does one (as the wiki page suggests) show that these sequences are Cauchy? Does one have to provide a direct proof (which I was having a hard time with due to the recursive definition) or use the fact that $A_1 \leq A_2\leq ...\leq B_2 \leq B_1$ and $|A_n -B_n|$ tends to 0?