Since is Cauchy, there exist N such that if m,n are both larger than N, then . Let M be the largest of for m and n both less than or equal to N+1.
Now, show that for all n and m.
(Consider the cases: (a) n and m both less than N, (b) n and m both larger than N, (c) one of m or n less than N and the other larger.)
Finally show that for all n, |a_n|< |a_1|+ M+1.