
Cauchy Sequence
Dear Colleagues,
Could you please help me in solving the following problem:
If $\displaystyle (x_{n})$ is a Cauchy sequence in the metric space $\displaystyle (X,d)$ and $\displaystyle (y_{n})$ in $\displaystyle X$ such that $\displaystyle d((x_{n}),(y_{n}) \longrightarrow 0 $as $\displaystyle n\longrightarrow \infty$then show that$\displaystyle (y_{n})$ is Cauchy in $\displaystyle X $.

Have you tried drawing a 2d diagram? (they can be great models for metric spaces) Remember since the $\displaystyle x_n$'s are Cauchy you can find an epsilon ball that the tail never leaves.
Now since the $\displaystyle y_n$'s get close to the $\displaystyle x_n$'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.

Note that for all positive number $\displaystyle \epsilon $, you can find $\displaystyle N \in \mathbb {N} $ such that $\displaystyle d(x_n,x_m) < \frac { \epsilon }{3} $, likewise for $\displaystyle d(x_n,y_m) $
Now, consider $\displaystyle 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) $