Prove, by definition, that if then .
It should be a basic proof but I'm somehow stuck. I was obviously thinking:
but I'm not sure how to proceed.
Any help would be appreciated.
You are given an ε > 0 and you need to make |aₙ - L|⋅|aₙ + L| < ε by choosing n. You can make |aₙ - L| arbitrarily small by choosing n big enough. Also, you can make |aₙ + L| arbitrarily close to 2L by choosing n big enough. In particular, you can make |aₙ + L| < 3L for some n and onward. Assuming you have done this, how small do you need to make |aₙ - L| so that |aₙ - L|⋅|aₙ + L| becomes smaller than ε?