Thread: Find an "interesting" Cauchy sequence.

For a rational number write it as where and are integers that have no common factors, and let Define Let

Prove for all rational

Now since and with equality only if Write an interesting Cauchy sequence in this metric.

Originally Posted by cgiulz

For a rational number write it as where and are integers that have no common factors, and let Define Let

Prove for all rational

Now since and with equality only if Write an interesting Cauchy sequence in this metric.

For positive integers m,n we have that m > n <==> 2^(-m) < 2^(-n) , so

|x + y|_2 <= max{|x|_2 , |y|_2} <= |x|_2 + |y|_2 and from here the

triangle inequality follows.

As for an interesting Cauchy sequence: what about the sequence

{2^n} = {2, 4, 8, 16,...} ?

Tonio