Proving existence of limit

Question

What is a way to prove the existence of a limit of the difference of two Cauchy sequences? What is a general definition that can be used to prove that a limit exists?

Answer 1

Working over $\mathbb{R}$, let $(x_n)$ and $(y_n)$ be two Cauchy sequences. Since $\mathbb{R}$ is complete, both sequences have limits, say $x$ and $y$, respectively. Now, consider the sequence $(x_n - y_n)$. Let $\varepsilon > 0$. We then have the following: \begin{equation} |(x_n - y_n) - (x-y)| = |(x_n - x) + (y - y_n)| \leq |x_n - x| + |y_n - y| \end{equation} Since both sequences converge, there exists an $N \in \mathbb{N}$ such that if $n \geq N$, we have $|x_n - x|, |y_n - y| < \varepsilon/2$, and therefore $|x_n - x| + |y_n - y| < \varepsilon$ for sufficiently large $n$. We may conclude that $|(x_n - y_n) - (x - y)|< \varepsilon$ for sufficiently large $n$. Hence, the sequence of differences is convergent, and thus Cauchy, in $\mathbb{R}$.

edit: we have of course shown that the limit of the sequence of differences is the difference of the limits, $x-y$.

Answer 2

The difference of two Cauchy sequences can be shown to be Cauchy. In $\mathbb R$, every Cauchy sequence converges.