I'm really confused on how epsilon is chosen (i.e. the red arrows in the picture), and how those inequalities are worked out.
In this example, is there a simpler value of epsilon that could be chosen?
I (think I) understand how it works - you're proving that once you reach a certain point in the sequence (N) |an-l|, the difference between a term in the sequence and its limit, will always be smaller than an arbitrarily chosen small number.
ok, in this example is there anything else fairly obvious that epsilon could have been, or is this the only real solution?
You seem a little confused; your aim is find N given a particular value of ϵ.
So I suspect you mean is there anything else obvous that N could be (other than 9/2ϵ.
And the answer is yes. All you need to do is find an N that works (i.e.N such that for every n > N we have ∣an−a∣<ϵ). So obviously if N works, anything bigger than N will work too.
so 100000/ϵ would work, for example.
Note in particular that there's no requirement for N to be the smallest value that will work. It's important to understand this, since It's often impossibly hard to find the smallest value that will work.