I have the following sequence: $\{u_{n}\}$ wich goes as follows:
If $n = 2k-1$, then: $u_{n}= x_{k}$
And if $ n=2k$, then: $u_{n} = y_{k}$
Both $\{y_{n}\}$ and $\{x_{n}\}$ are Cauchy. I need help proving that $\{u_{n}\}$ is cauchy if and only if:
$$\lim_{n \to \infty} \rho(x_{n},y_{n}) = 0$$
For the first half I used the definition of cauchy sequence: for every $\epsilon > 0$ there exist a natural number $N$ such that for every $n,m > N$:
$$\Vert u_{n} - u_{m} \Vert < \epsilon$$
Then, I considered the case where $n = 2k -1$. Which implies that
$$u_{n} = x_{k}$$
After that, I set $m = n + 1$. Or in other words, $m = 2k$. Which would, on the other hand, imply that $$u_{m} = y_{k}$$
Then I simply plugged $x_{k}$ and $y_{k}$ inside the norm. $$\Vert x_{k} - y_{k} \Vert < \epsilon$$
Finally concluding with
$$\lim_{k \to \infty} \rho(x_{k}, y_{k}) = 0$$
That would be the first half of the proof, but I don't know how to go on the opposite way. I've been struggling with this particular problem for a while now. Any tips? Thanks a lot!