If (Xsubn) and (Ysubn) are Cauchy Sequences, show that (Xsubn + Ysubn) and (Xsubn*Ysubn) are Cauchy Sequences. I am asked to prove this using the definition of a Cauchy Sequence only. Any help is much appreciated.
i will do the first (since it is easier ) you do the second
since is a Cauchy sequence, we have that for all there exists , such that implies
similarly, we have so that implies
Now, choose . then implies
so that is Cauchy
(note that the follows by the triangle inequality)
now do the second. use the defintion as i did