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porroadventum
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Homework Statement
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.
Homework 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
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.