A sequence {an} in a metric space is called a Cauchy sequence iff for any (positive) ε, there exists some N=Nε such that d(an,am) < ε for all n,m ≥ N.

If the metric space is complete, then the limit of the sequence lim an exists. Proving a sequence is a Cauchy sequence can be easier than showing its limit directly (because we don't need to produce the actual limit!). This is an advantage of working in a complete setting. See the Baire category theorem for a more stunning advantage...

A sequence {a_n} is a Cauchy Sequence if and only if for every E>0 there exists a positive integer N such that if m,n >= N then |a_n - a_m|> E (Read E as epsilon)

Basically a sequence, {a1,a2,a3,a4,a5...an}, is Cauchy if the terms in the sequence begin to 'get closer and closer together' as you get farther away from a1.

An example of a Cauchy Sequence would be:
{1/n}
The terms of this sequence are {1, 0.5, 0.33, 0.25, 0.2, 0.17...,.01 ,...1/n} This sequence is Cauchy because as n goes to infinity the terms get closer and closer together. This process of 'getting closer and closer together' is also known as convergence.

Math definition: A Cauchy sequence is some sequence of numbers, call it an, such that for every E > 0, there is a positive integer N such that if j and k are greater than N, then |aj-ak| < E.

In plain English: Say you have an infinite sequence of numbers. Normally this sequence would be defined by a mathematical expression, such as an = 1/n = {1, 0.5, 0.333..., 0.25, 0.2, 0.166..., etc.}.

Now pick some positive number E, as small as you like. 0.1, 0.01, 0.001, whatever. (Just don't be a smart aleck and try dividing by infinity.)

In order for this sequence to be a Cauchy sequence (or just "be Cauchy", as the math teachers inevitably end up saying), there must be some point in your number sequence such that any two numbers after that point differ by less than E. If E is very small, N may have to be very large, but as long as it exists, your sequence is Cauchy.

For example: Let's say we use the sequence an = 1/n and E=0.1. Then a good value for N would be 10, since a10 = 1/10 = 0.1 and any two numbers in the sequence after that must be between 0.1 and zero--and therefore their difference will be between 0.1 and zero, too. If E=0.01, N=100, and so on and so forth. No matter how small E gets, the sequence will (eventually) squeeze under it.

So what?: A Cauchy sequence is technically different from a convergent sequence in that you don't have to know what number it converges to (which can be handy if that number is irrational--see xriso's writeup below). However, it can be proven that where the real numbers are concerned, all Cauchy sequences are convergent sequences and vice versa. This means that Cauchy sequences can be used to determine whether a real-number function is continuous and makes them an essential piece of calculus and real analysis proofs.

So, you might ask "What this definition is useful for?"

Cauchy sequences are a way of constructing the reals out of the rationals.

Now, let's forget all we know about the real numbers, so that the best things we can work with are the rationals. Numbers like 5/4, -2, 0.0006, 4367582828365978387934. Say we want to solve the equation x^2 = 2. Of course, one answer is x = the square root of 2, but it is not a rational number. We have a pretty good idea of where it should be: something higher than 1, but less than 2. We can delve a little deeper, to restrict it between 1.4 and 1.5. Deeper still: 1.41 and 1.42. If we try hard enough, we can make its square as close to 2 as we want, but not exactly at 2. Let's construct a sequence that keeps adding one more decimal figure of the "square root of two": {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...}. Note that every member of this sequence is a rational number.

If we try to find out which rational number this sequence is converging to, then we're out of luck because the sequence does not converge to any such number. This is where Cauchy sequences come to the rescue, because this sequence is in fact Cauchy.

The result: We have a sequence of elements whose *squares* are converging to 2 (that's how we constructed it), and the sequence itself keeps getting tighter and tighter (since it's a Cauchy sequence).

So, we gather up all other sequences of elements whose squares converge to 2, and put them in a set. That set is the real number "square root of 2".

Another sequence in the set is {2, 1.5, 1.42, 1.415, 1.4143, 1.41422, ...}.

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