How do you prove that a sequence converges?

Step 1: Find LL, the limit of the sequence

First, figure out what number LL you think the sequence converges to.

Step 2: Choose N\textcolor{#1d4ed8}{N} for an arbitrary ε\varepsilon

Let ε>0\varepsilon > 0. Choose N=\textcolor{#1d4ed8}{N = } .

Choose so that beyond N\textcolor{#1d4ed8}{N}, all the terms are within ε\varepsilon of LL. (Because that's what we're going to prove next.)

Step 3: Prove that your choice of N\textcolor{#1d4ed8}{N} works

Finally, you need to prove that beyond your N\textcolor{#1d4ed8}{N}, all the sequence terms are within ε\varepsilon of LL.