# Notation for sequences

I am trying to write a small article, and I just want to know how would be a good way to present the maths I have written so that it looks professional.

I am trying to define a sequence $x_n$ of real numbers. So what I wrote in my article is:

Let $x_n$, $n \in \mathbb{N}$ be a sequence of real numbers.

However, it does not look very professional. How would I write the above sentence into something that looks professional? Note: I need to include $n \in \mathbb{N}$ in my sentence, so I think thats where my trouble is as $x_n$, $n \in \mathbb{N}$ seems a bit messy.

Thanks.

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I prefer to present a sequence as $(x_n)_{n\in \mathbb{N}}$ (the parentheses are used to mean that a sequence is "ordered" in some way, just like finite $n$-tuples do). Other people use also $\{x_n\}_{n\in \mathbb{N}}$ (but I don't like this symbol, because curly brackets always denote sets and sets are not necessarily "ordered"). –  Pacciu Mar 3 '12 at 15:29
Is it an infinite sequence? Then maybe let $x_n$, where $n$ ranges over $\mathbb{N}$, be real numbers. Using more symbols is not necessarily more professional. –  André Nicolas Mar 3 '12 at 15:31
Thanks for your help guys! Yes it is an infinite sequence. –  Gary Mar 3 '12 at 15:47
Just to add to the previous answers, in the Z formal specification language, curly brackets denote a set and angle brackets denote a sequence. copied from this link (3rd Paragraph) –  Alma Rahat May 9 '13 at 14:40

I write my sequences as $\langle x_n \mid n \in \mathbb{N} \rangle$. Looks pretty cool I think. So you could say let $\langle x_n \mid n \in \mathbb{N} \rangle$ be a sequence in $\mathbb{R}$.

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Notation $\langle \cdot \mid \cdot \rangle$ is used by physicists to denote inner product in the sequences space $\ell^2$; also, most mathematicians use the angular brackets $\langle \cdot ,\cdot \rangle$ to denote the inner product in a Hilbert space (hence also in $\ell^2$). Hence, IMHO, the notation you use can be very ambiguous. –  Pacciu Mar 3 '12 at 17:41
I probably use this more since this is the standard notation for sequences in set theory. –  Paul Slevin Mar 3 '12 at 18:07
I think it's not too ambiguous in this case. Probably no one would denote an inner product with $n \in \mathbb{N}$ in one of the sides of the angular brackets. –  user1620696 May 9 '13 at 15:05
Big angle brackets is how Lamport denotes sequences in the TLA+ ecosystem. Of course he goes through great pains to make it clear a sequence is formally a function $x: \mathbb{N}\rightarrow S$ –  Michael Deardeuff Sep 13 '13 at 22:26

The notation $\{x_n\}_{n\in\mathbb{N} }$ is appropriate because a sequence is formally a function $x: \mathbb{N}\rightarrow S$ that maps natural numbers to elements of the set $S$ (codomain). The index $n$ denotes the argument of function $x$. The first notation corresponds to a set of numbers $\{x_n\}$ (the curly brackets do denote a set) indexed by the natural numbers.

Note also that the space of all real-valued sequences can be denoted as $\mathbb{R}^\mathbb{N}$, i.e., the set of all functions from the natural numbers $\mathbb{N}$ to the real numbers $\mathbb{R}$. See also this discussion: What does it mean when a set is the exponent?

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I write my sequences as and don't mention n in natural no. As by very definition of sequence these are functions with domain natural no. I avoid {xn} to make difference between range of sequence and sequence itself e.g. <1, -1, 1, -1...> is sequence with range {1,-1}

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