Intro to Real Analysis: Question on a completed proof.

**We had to prove this:**

Let http://www.cramster.com/Answer-Board...6875004376.gifand http://www.cramster.com/Answer-Board...2187502633.gif be sequences of real numbers. If http://www.cramster.com/Answer-Board...8437506419.gif is bounded and http://www.cramster.com/Answer-Board...4687509666.gif http://www.cramster.com/Answer-Board...7812505844.gif= 0,

then http://www.cramster.com/Answer-Board...4687509666.gif http://www.cramster.com/Answer-Board...2500008978.gif= 0.

(The fact that http://www.cramster.com/Answer-Board...2187502633.gif is bounded is crucial.)

I found this:

http://i33.tinypic.com/2yxoxl2.jpg

My question is where the red arrow is pointing and the red boxed part:

How do I know how to choose 1/M for ε?

In this case we want to prove |$\displaystyle a_n$$\displaystyle b_n$| < ε

so I need to choose any ε value right? How do I know to choose 1/M?

Is there algebra I can do that will lead me to choosing 1/M?

I think I understand the rest of the proof up to that point where the red arrow is pointing to.

Any help, suggestions are appreciated. Thanks for your time!