Let $(x_n)_{n∈ℕ}$ and $(y_n)_{n∈ℕ}$ be Cauchy sequences of real numbers.
Show, without using the Cauchy Criterion, that if $z_n=x_n+y_n$, then $(z_n)_{n∈ℕ}$ is a Cauchy sequence of real numbers.
Here's my attempt at a proof:
Let $(x_n)$ and $(y_n)$ be Cauchy sequences. Let $(z_n)$ be a sequence and let $z_n=x_n+y_n$.
Since $(x_n)$ and ($y_n)$ are Cauchy, $∃N∈ℕ$ such that $|x_n-x_m|<ε/2$ and $|y_n-y_m|<ε/2$ for $n,m≥N$.
Let $n,m≥N$ and let $z_n,z_m∈(z_n)$. Then, $|z_n-z_m|=|x_n-x_m|+|y_n-y_m| <ε/2+ε/2=ε.$ Therefore, $|z_n-z_m|<ε$ for all $n,m≥N$ and hence, $(z_n)$ is a Cauchy sequence of real numbers.
Is this correct? Any input is appreciated.
Thanks.