Re: Cauchy Sequence proof

Because this is a Cauchy sequence, you know that for some N, if n,m> N, then $\displaystyle |a_n- a_m|< 1$. In particular, if $\displaystyle a_n> a_{N+1}$, we have $\displaystyle |a_n- a_{N+1}|= a_n- a_{N+1}< 1$ so that $\displaystyle a_n< a_{N+1}+ 1$. So $\displaystyle a_{N+1}+ 1$ is an upper bound for all terms with n> N.

The part you are asking about, where they look at $\displaystyle |a_1|$, up to $\displaystyle |a_N|$ is the easy part- those together with the bound $\displaystyle a_{N+1}+ 1$ form a **finite** set. And a finite set of numbers always contains a "largest member" which must be an upper bound on the entire sequence.