How do you prove that a sequence converges?
Step 1: Find L, the limit of the sequence
First, figure out what number L you think the sequence converges to.
Step 2: Choose N for an arbitrary ε
Let ε>0. Choose N= .
Choose so that beyond N, all the terms are within ε of L. (Because that's what we're going to prove next.)
Step 3: Prove that your choice of N works
Finally, you need to prove that beyond your N, all the sequence terms are within ε of L.