We had to prove this:

Let and be sequences of real numbers. If is bounded and = 0,

then = 0.

(The fact that is bounded is crucial.)

I found this:

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 | | < 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!