# Cauchy Sequence

• Apr 7th 2011, 05:33 AM
raed
Cauchy Sequence
Dear Colleagues,

If $(x_{n})$ is a Cauchy sequence in the metric space $(X,d)$ and $(y_{n})$ in $X$ such that $d((x_{n}),(y_{n}) \longrightarrow 0$as $n\longrightarrow \infty$then show that $(y_{n})$ is Cauchy in $X$.
• Apr 7th 2011, 06:14 AM
TheEmptySet
Have you tried drawing a 2d diagram? (they can be great models for metric spaces) Remember since the $x_n$'s are Cauchy you can find an epsilon ball that the tail never leaves.

Now since the $y_n$'s get close to the $x_n$'s can find another N such that this distance also falls in another epsilon ball.

As I said draw a picture and what fraction of epsilon you need.

Hint you will need 3 balls.
• Apr 7th 2011, 09:19 AM
Note that for all positive number $\epsilon$, you can find $N \in \mathbb {N}$ such that $d(x_n,x_m) < \frac { \epsilon }{3}$, likewise for $d(x_n,y_m)$
Now, consider $d(y_n,y_m) \leq d(y_n,x_n) + d(x_n,y_m) \leq d(y_n,x_n) + d(x_n,x_m)+d(x_m,y_m)$