Because this is a Cauchy sequence, you know that for some N, if n,m> N, then . In particular, if , we have so that . So is an upper bound for all terms with n> N.
The part you are asking about, where they look at , up to is the easy part- those together with the bound 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.