Hello!

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

The question is, given two Cauchy sequences and , in a metric space X (with a metric "d"), we want to show that

converges.

What I've got so far:

We are given and are Cauchy, so, by definition, given , such that and whenever .

So, by the triangle inequality we have

then it follows thats:

but,

and

so

so when

that fulfils the definition of Cauchy, so

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.