Show that {x n } is a Cauchy sequence. Show that theconverse is false. Solution

Question

Show that {xn} is a Cauchy sequence. Show that theconverse is false.

Explanation / Answer

Take that becuase ? converges... then the sequence of partialsums converge, and the residue approches 0. This means, the sum ?_k, with wich I denote ?_kd(x_n, x_{n+1}) with n=k to infty (this are the residues),approach 0. Lets take then that d(x_k, x_k+1) = ?_k - ?_k+1 >=0.We want to see this approaches 0. One more thing to note, thesum is monotone increasing, because the distance funtion is >= 0always, so the terms are positive. This means the residues are alsomonotone decreasing. With this, take >0. Then there is N such thatif k>N, ?_k < . But then d(x_k, x_k+1) ?_k- ?_k+1 < ?_k < , for all k>N. Becuase is small as we can take, and because of monotonicity of ?_k,then the distances tend to 0. We do the same thing, taking the substraction, to see the distancesbetween two different x_m, x_n, with n,m>k. Now taking d(x_m, x_n)

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