1. The problem statement, all variables and given/known data Show that the sequence (xn)n[itex]\in[/itex]N [itex]\in[/itex]Z given by xn = Ʃ from k=0 to n (7n) for all n [itex]\in[/itex] N is a cauchy sequence for the 7 adic metric. 2. Relevant equations In a metric space (X,dx) a sequence (xn)n[itex]\in[/itex]N in X is a cauchy sequence if for all ε> 0 there exists some M[itex]\in[/itex]N such that dx(xn,xm)<ε for all m,n ≥ M. the 7-adic metric is defined as follows: p(m,n)= 1/(the largest power of 7 dividing m-n) if m[itex]\neq[/itex]n or 0 if m=n 3. The attempt at a solution I am struggling with proving sequences are cauchy because I am not sure how to go about finding the 'M'? I am not even sure how to start the question apart from assuming m<n. Just a hint at how to start it or how to approach the question would be appreciated. Assuming m<n am I able to write d7(xn,xm)=Ʃ from k=m+1 to n (7n)? Thank you.