i want to take a good sub-sequence out of a Cauchy sequence.
Let $(x_n)$ be cauchy sequence , then there exist sub sequence $(x_{n_k})$ where $\sum d(x_{n_k},x_{n_{k+1}}) < \infty$ is satisfied.
My sketch of proof
$\text{Since } (x_n) \text{ is a Cauchy sequence, we can take repeatedly } \\ N_1 : n>m≥N_1 \Rightarrow d(x_n,x_m)< (\frac{1}{2})^1,\\ N_2 (>N_1): n>m≥N_2 \Rightarrow d(x_n,x_m)< (\frac{1}{2})^{1/2},\\ N_3(>N_2): n>m≥N_3 \Rightarrow d(x_n,x_m)< (\frac{1}{2})^{1/4},\ldots$
After all, the sub-sequence $x_1, x_2, \ldots ,x_{N_1},x_{N_2},x_{N_3},\ldots$ satisfies the desired condition. Because it holds $d(x_{N_1},x_{N_2})<1/2, d(x_{N_2},x_{N_3})<1/4,...$
Is there something wrong?
Almost. It should simply be $x_{N_1},x_{N_2},x_{N_3},\ldots$; actually, I am not really sure about what you meant when you wrote $x_1,x_2,\ldots,x_{N_1},x_{N_2},x_{N_3},\ldots$