Definition 2.0.1.
A real sequence (or sequence in \(\mathbb{R}\) or sequence of real numbers) is a function \(X:\mathbb{N}\to\mathbb{R}\text{.}\)
Chapter 2 Real sequences
As I mentioned at the beginning of these notes, the point of analysis is to study the behavior of functions. We’ll start with a special kind of function.
For now we’re going to save breath and ink by referring to real sequences as just sequences, but you should be aware that we’ll encounter other kinds of sequences later on.
Convention 2.0.2.
We usually depart from standard function notation, and write things like \(X=\left(x_n\right)_{n\in\mathbb{N}}\text{,}\) where \(x_n\) means \(X(n)\text{.}\)
Example 2.0.3.
Here are some sequences:
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\(\displaystyle X(n)=n^2\)
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\(\displaystyle x_n=1+\frac{1}{n}\)
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\(\displaystyle \left(\frac{1}{k^3+1}\right)_{k\in\mathbb{N}}\)
It’s very important to distinguish between the image or trace of a sequence, which is a set of real numbers, and the sequence itself, which is a function.
Example 2.0.4.
The constant sequence \(X(n)=3\text{,}\) aka \((3)_{n\in\mathbb{N}}\text{,}\) is infinite; whereas its image \(\left\{3\right\}\) is a singleton.
The main difference is that a set is an unordered collection of elements, whereas the terms of a sequence come in a definite (and infinite!) order.
