Cauchy sequences

• October 26th 2008, 01:34 AM
terr13
Cauchy sequences
Suppose $x_n$ is a Cauchy sequence in a metric spare X, and some subsequence $x_k$ converges to a point $p \in X$. Show the full series converges to p.

Is this correct?

First prove convergence of $x_n$. Find an integer N such that $\forall a,b \geq N, d(x_a, x_b) \leq \epsilon/2$. Now set N so that $d(x_k,x) \geq \epsilon /2$ for $x_k \geq N$. Now replace $x_b$with $x_k$, then $\epsilon \geq d(x_a,x_k) +d(x_k,p) \geq d(x_a,x)$

Now prove that the sequence goes to p. Assume x $_n \rightarrow q$ and $d(p,q) \leq \epsilon$ for $\epsilon \geq 0$. Pick an integer N such that $d(x_n,x_{n+1}) \geq \epsilon$ This cannot be possible since it would violate the fact that $x_n$ is Cauchy, so $d(p,q)=0 \,\,and \,\,p=q$
• October 26th 2008, 05:56 AM
kalagota
Quote:

Originally Posted by terr13
Suppose $x_n$ is a Cauchy sequence in a metric spare X, and some subsequence $x_k$ converges to a point $p \in X$. Show the full series converges to p.

Is this correct?

First prove convergence of $x_n$. Find an integer N such that $\forall a,b \geq N, d(x_a, x_b) \leq \epsilon/2$. Now set N so that $d(x_k,x) \geq \epsilon /2$ for $x_k \geq N$. Now replace $x_b$with $x_k$, then $\epsilon \geq d(x_a,x_k) +d(x_k,p) \geq d(x_a,x)$

Now prove that the sequence goes to p. Assume x $_n \rightarrow q$ and $d(p,q) \leq \epsilon$ for $\epsilon \geq 0$. Pick an integer N such that $d(x_n,x_{n+1}) \geq \epsilon$ This cannot be possible since it would violate the fact that $x_n$ is Cauchy, so $d(p,q)=0 \,\,and \,\,p=q$

maybe this would help..
$d(x_n, p) \leq d(x_n, x_k) + d(x_k,p)$