Alright, Cauchy sequences and convergence…or at least try to make sense of them without falling asleep. So first things first: a sequence is just a list of numbers that goes on forever (or at least as far as we can tell). For example, 1, 2, 3, 4, 5…is a sequence. A Cauchy sequence is a special kind of sequence where the terms get closer and closer to each other as you go along.

Here’s an example: let’s say our sequence looks like this: 1/2, 3/4, 7/8, 15/16…and so on. If we keep going out further, the numbers get really close together! In fact, if you take any two terms in that sequence (let’s call them a and b), no matter how far apart they are, there will always be another term c somewhere between them that is even closer to both of them than either one is.

That’s what makes this a Cauchy sequence: it gets arbitrarily close to any given pair of terms in the sequence. And if you can find a Cauchy sequence like that, then there’s a good chance (but not necessarily) that the sequence converges to some number.

Now, convergence is where things get tricky. A sequence converges if it gets closer and closer to some fixed value as we go along. For example, 1/2, 3/4, 7/8…converges to 1 (if you keep going out far enough). But not all Cauchy sequences converge! In fact, there are plenty of Cauchy sequences that just wander around in a circle or something and never settle down.

So how do we know if a sequence converges? Well, one way is to use the epsilon-delta definition: if for any given number (let’s call it ε) there exists some other number (let’s call it δ) such that whenever our terms are more than δ apart from each other, they will be less than ε apart from their limit…then we can say the sequence converges.

That sounds like a mouthful, but basically what this means is that if you pick any number (like 0.1 or something) and then choose some other number (let’s call it d) that’s smaller than your first number, there will always be some point in the sequence where all of the terms are within d of each other…and they’ll also be within ε of their limit.

So if you can find a Cauchy sequence like that (where every pair of terms is getting closer and closer to each other), then it might converge! But there’s no guarantee, because not all Cauchy sequences converge…and sometimes they just wander around in circles or something.

But hey, at least now you know what a Cauchy sequence is and how convergence works (sort of). And if that doesn’t help you pass your math class, then I don’t know what will!