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 $\textcolor{#1d4ed8}{N}$ for an arbitrary $\varepsilon$

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

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

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

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