# Math Help - cauchy

1. ## cauchy

Suppose that {xm} is a Cauchy sequence, and that 􏰫xm(k)􏰬 is a subsequence of {xm} which converges to some point p. Prove that {xm} itself must also converge to p.

2. Let $\epsilon>0$

Since $\{x_n\}$ is Cauchy, we know for some $N>0 \;\; \forall\; n,m>N, \; d(x_n,x_m)<\tfrac{\epsilon}{2}$.

Since $\{x_{n_k}\}$ converges to $p$ we also know for some $M>0 \;\; \forall n_i,m>M, \; d(x_{n_i},p)<\tfrac{\epsilon}{2} \text{ and } d(x_{n_i},x_m)<\tfrac{\epsilon}{2}$.

By the triangle inequality $d(x_m,p)<\epsilon$.