# 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 $L$.