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.