Prove that the sum of two Cauchy sequences is Cauchy without using:
Let {x_n} be a sequence of real numbers. Then {x_n} ic Cauchy iff {x_n} converges.
Bored.
Let and be Cauchy sequences.
Define . We show that is Cauchy.
Since is Cauchy, for all , there exists an such that implies .
Similarly, since is Cauchy, we can find an such that implies
Now, fix such an , and choose . Then implies that:
Thus, is a Cauchy sequence
QED.