1. ## Cauchy sequences

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

Is this correct?

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

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

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

Is this correct?

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

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