Sequences occur everywhere in our daily life. Some of the examples include sensor data, stock market quotes, speech signals, and many more. A sequence is a collection of elements where each element is indexed. Repetitions are allowed in this case, which means any element can reappear in a given sequence. If we look closely, we can see that sequences are rich in information. In theory, we can design sequences with amazing characteristics and study them. This allows us to approximate real world processes using these sequences so that we can estimate what’s going to happen in the future. Cauchy sequence is one such sequence that’s very fundamental to a lot of fields. Let’s dig deeper and see why it’s relevant, shall we?
What is a Cauchy sequence?
A Cauchy sequence is a sequence whose terms become arbitrarily close to each other as it progresses. An important thing to note is that it’s not sufficient a given element gets closer to the preceding element. It’s important that all the elements of the sequence after a particular point become close to each other. Going by this definition, the elements of the harmonic series wouldn’t form a Cauchy sequence.
For example, consider the following figure of a sine wave:
We can see that the values are not getting close to each other. It keeps oscillating between +1 and -1. Now consider the following figure:
We can see that the values are getting close to each other as the wave progresses. This is an example of a Cauchy sequence. The good thing about Cauchy sequences is that they have some interesting properties. This makes them very well suited to model real world processes. Let’s see what those are.
What’s so special about Cauchy sequences?
We can use Cauchy sequences analysis when we are trying to prove that a given sequence converges to something but don’t know what that value is. As we saw in the preceding section, all Cauchy sequences are bounded. It means that once we cross a particular point in the sequence, there is an upper bound and a lower bound on all the elements. Every element after this point will be bounded between these two points.
Any convergent sequence is also a Cauchy sequence, but not all Cauchy sequences are convergent. Proving that is beyond the scope of this blog post. But a quick way to understand it would be that the convergent value must also belong to the given domain.
Let’s consider the example of rational numbers (any number of the form p/q where p and q are integers and q is not 0). Consider the following sequence:
f(n) = (1 + 1/n)n
where n is greater than 0.
If you construct the sequence, you can see that it’s a sequence of rationals. But the limiting value of the above sequence is ‘e’, which is an irrational number that doesn’t belong to our rational number domain.
This gives rise to another interesting property. A Cauchy sequence in the real numbers domain is convergent. This is used heavily in the real world because we encounter real numbers most of the time.
How do we use it in the real world?
Cauchy sequences have amazing properties that can be used to understand the behavior of a system as time progresses. They are heavily used in fields like satellite design, manufacturing, construction, treatment plants, and so on.
Let’s say we want to analyze how a structure is going to respond to weather conditions in the next couple of decades. Based on historical data, if we know that this can be modeled as a Cauchy sequence, we can extract a lot of insights. The goal is to model the system and make predictions about how this structure is going to react under various conditions. If we know that a given physical process can be described using a Cauchy sequence, we know that the process will converge to a single point.