Problem:

If {$\displaystyle a_n$} and {$\displaystyle b_n$} are Cauchy sequences in $\displaystyle \mathbb{R}$, prove that {$\displaystyle a_n + b_n$} and {$\displaystyle a_nb_n$} are also Cauchy sequences.

You cannot use the Theorem 2.6.4 which states, "Every Cauchy sequence of [tex]\mathbb{R}/MATH] converges.

My work:

We can't use Theorem 2.6.4, but Theorem 2.6.2 says, "Every convergent sequence of $\displaystyle \mathbb{R}$ is a Cauchy sequence." So in order for me to prove {$\displaystyle a_n + b_n$} and {$\displaystyle a_nb_n$} are also Cauchy sequences, I need to prove that {$\displaystyle a_n + b_n$} and {$\displaystyle a_nb_n$} converge.

I found this:

But our professor told us to prove this claim directly by using $\displaystyle \epsilon$, and think that proof above uses the theorem 2.6.4 which we are not allowed to use.

My work so far:

Assume {$\displaystyle a_n$} and {$\displaystyle b_n$} are Cauchy sequences in $\displaystyle \mathbb{R}$. Thus $\displaystyle \forall$ $\displaystyle \epsilon$ > 0, $\displaystyle \exists$ $\displaystyle n_o$ $\displaystyle \in$ $\displaystyle \mathbb{N}$ such that $\displaystyle |a_n - a_m|$ < $\displaystyle \epsilon$ $\displaystyle \forall$ $\displaystyle n,m \geq$ $\displaystyle n_o$. And also, $\displaystyle \forall$ $\displaystyle \epsilon$ > 0, $\displaystyle \exists$ $\displaystyle k_o$ $\displaystyle \in$ $\displaystyle \mathbb{N}$ such that $\displaystyle |b_v - b_w|$ < $\displaystyle \epsilon$ $\displaystyle \forall$ $\displaystyle v,w \geq$ $\displaystyle k_o$.

Let $\displaystyle \epsilon$ > 0. Since $\displaystyle |a_n - a_m|$ < $\displaystyle \frac{\epsilon}{2}$, also $\displaystyle |b_v - b_w|$ < $\displaystyle \frac{\epsilon}{2}$. Thus we have $\displaystyle |a_n - a_m + b_v - b_w|$ $\displaystyle \leq$ $\displaystyle |a_n - a_m|$ + $\displaystyle |b_v - b_w|$ < $\displaystyle \frac{\epsilon}{2}$ + $\displaystyle \frac{\epsilon}{2}$ = $\displaystyle \epsilon$.

This is where I got stuck/confused: How to prove {$\displaystyle a_nb_n$} converges using epsilon because when I multiply $\displaystyle |a_n - a_m|$ $\displaystyle |b_v - b_w|$, I get lost because I'm seeing too many different subscripts . How do I go about finishing the proof using epsilons to proof convergence so I can say the sequences are Cauchy sequences?

Any help, suggestions, corrections, and tips are greatly appreciated.

Thank you for your time.