Hello!

My question is one from Baby Rudin... Chapter 3, Question 23.

The question is, given two Cauchy sequences $\displaystyle \{p_n\}$ and $\displaystyle \{q_n\}$, in a metric space X (with a metric "d"), we want to show that

$\displaystyle \{d(p_n, q_n)\}$ converges.

What I've got so far:

We are given $\displaystyle \{p_n\}$ and $\displaystyle \{q_n\}$ are Cauchy, so, by definition, given $\displaystyle \epsilon > 0$, $\displaystyle \exists N$ such that $\displaystyle d(p_n, p_m) < \epsilon$ and $\displaystyle d(q_n, q_m) < \epsilon$ whenever $\displaystyle n, m > N$.

So, by the triangle inequality we have

$\displaystyle d(p_n, q_n) \leq d(p_n, p_m) + d(p_m, q_m) + d(q_m, q_n)$

then it follows thats:

$\displaystyle |d(p_n, q_n) - d(p_m, q_m)| \leq d(p_n, p_m) + d(q_m, q_n)$

but, $\displaystyle d(p_n, p_m) < \epsilon$

and $\displaystyle d(q_m, q_n) < \epsilon$

so

$\displaystyle d(p_n, p_m) + d(q_m, q_n) < 2\epsilon$

so $\displaystyle |d(p_n, q_n) - d(p_m, q_m)| < 2\epsilon$ when $\displaystyle n > N$

that fulfils the definition of Cauchy, so

$\displaystyle \{d(p_n, q_n)\}$ is Cauchy.

but... we don't neccecarily know it converges, right?

because X is a general metric space... we don't know if Cauchy sequences converge in a general metric space... they only converge in complete metric spaces.

Any help much appriciated!! Thank you.