# Does the notation $\{X_n\}_{n \geq 1}$ always imply that it is an *infinite* sequence?

In probability theory I have come across $\{X_n\}_{n \geq 1}$ and $(X_n)_{n \geq 1}^{+\infty}$ to signify a sequence of random variables $X_1,X_2,...,X_n,...$ However is $\{X_n\}_{n \geq 1}$ always implied to be infinite, or is $\{X_n\}_{n \geq 1}^{+\infty}$ the correct notation? And how should the corresponding notation be written out (i.e. $X_1,X_2,...$ etc)?

• I've always seen $\lbrace X_n\rbrace_{n\geq 1}$ to imply infinite (Specifically, countable) sequences. – Mark Dec 7 '16 at 8:56

Often, in mathematics, a sequence is, per definition, infinite. So if you say that you have a sequence of real numbers, then it is often meant to be an infinite ordered list of real numbers. If you want, however, you could define all numbers above a certain value of the index to be zero. So you could have $(0,1,2,3,0,0,0,...)$ as an example. It can also be seen as a function $f:\mathbb{N}\to \mathbb{R}$, where the index set $\mathbb{N}$, in your sequence, thus serves as the domain of the function.

Also, $\{X_n\}_{n \geq 1}$ is meant to say the same thing as $\{X_n\}_{n = 1}^{\infty}$. The notation is similar to the one for series (infinite sums). You can often read $\sum_{n\geq 0}$ where you could write $\sum_{n=0}^{\infty}$.

EDIT 1: I have, however, seen definitions of a sequence as an ordered list of numbers, possibly infinite. One would then talk about finite and infinite sequences. At least within analysis, I believe it is more common to define them as infinite ordered lists. For reference, here is a definition from Walter Rudin - Principles of Mathematical Analysis:

"By a sequence, we mean a function $f$ defined on the set $J$ of all positive integers. If $f(n)=x_n$, for $n\in J$, it is customary to denote the sequence $f$ by the symbol $\{x_n\}$, or sometimes by $x_1,x_2,x_3,...$."

EDIT 2: As has been commented below by Did, the necessity of a sequence to be infinite might not be as universal as I thought (which was partly included in EDIT 1). It is however, in my opinion, very common. I want to note though, that the answer to your original question should still be a firm yes, i.e that $\{x_n\}_{n\geq 1}$ should refer to an infinite sequence. Read the comments below for some more discussion about this, but in conclusion, I think that your question has been answered.

However, I think the notation $(a_n)$ is more common. I mostly write that, or $(a_n)_{n=0}^{\infty}$.

Hope this clears things up.

• Thanks for the answer! On a side note, is there any difference then between using curly brackets $\{ X\}$ and parenthesis $(X)$? – litmus Dec 7 '16 at 9:12
• I have edited my answer above with an answer to this and an elaboration on some other things, so please read it again. But in short, no, it is just notation, and at least my professor (and my self), seem to prefer $(a_n)_{n=0}^{\infty}$. I think this is to emphasize that order is important here, since '{' are used for sets as well. But both notations appear. The limits are sometimes omitted if it is unimportant or clear from context. You would use an index to denote a sequence though, so write $\{x_n\}$ in that case, but as I mention, preferably $(x_n)$. – Christopher.L Dec 7 '16 at 9:18
• "Generally, in mathematics, a sequence is, per definition, always infinite" You think so? See first paragraph here or first paragraph there or... – Did Dec 7 '16 at 9:21
• @Did :Yes, I edited my answer before to include that I have seen other definitions, where they don't have to be infinite, that is true. You may be right in that it may be more common than I thought, that is also true. I therefore want to emphasize, that what I wrote is based on my own personal experience after 4 years of studies. The non-infinite sequences is something that I would call an n-tuple. But again, yes it might perhaps not be as universal as I thought. This is also partly why I included a reference from a highly respected book on analysis. – Christopher.L Dec 7 '16 at 9:30
• Also, to clear things up for litmus, the ambiguity discussed in the comments above does not really refer to your original question, where you asked if the notation $\{x_n\}_{n\geq 1}$ always imply that it is infinite. If you wanted to denote a finite sequence, you would write $\{x_n\}_{n=1}^{k}$, for some $k$ in that case. So at least there should be no ambiguity there. – Christopher.L Dec 7 '16 at 9:35