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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").2012-03-03
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    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.2012-03-03
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    Thanks for your help guys! Yes it is an infinite sequence.2012-03-03
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    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)](http://en.wikipedia.org/wiki/Bracket_%28mathematics%29)2013-05-09

2 Answers 2

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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.2012-03-03
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    I probably use this more since this is the standard notation for sequences in set theory.2012-03-03
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    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.2013-05-09
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    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$2013-09-13
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    @Michael Deardeuff: In TLA+, angle brackets denote tuples, which are finite sequences. For example, $\langle a, b \rangle$. In other words, they are "tuple constructor" syntax. They cannot be used to define infinite sequences, so they cannot be used to answer the OP. Instead, one has to define a function $f$ with $\mathrm{DOMAIN}\ f = \mathit{Nat}$.2016-12-10
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The best notation for functions that I have encountered so far is by Leslie Lamport, described in his book on the temporal logic of actions, TLA+ (see Sec.5.2 on p.48, and Sec.6.4 on p.69).

In axiomatic set theory as formalized in TLA+, a function is a set that satisfies certain axioms about relevant syntactic constructs. I have found the syntax exceptionally practical and precise:

  • $f \triangleq [x \in S \mapsto e(x)]$ defines a function that maps elements of the set $S$ to the expression $e(x)$. Example of function constructor syntax: $f \triangleq [x \in \{3, 4\} \mapsto x - 1]$

  • $\mathrm{DOMAIN}\ f$ is the domain of function $f$. If $f$ is defined as above, then $\mathrm{DOMAIN}\ f = \{3, 4\}$

  • $f[x]$ function application, is a known value only if $f$ is a function (an axiom schema tells us that whatever is defined using the function constructor syntax from above is a function). Otherwise, it is some unspecified value (value is synonym for set in the set theory underlying TLA+).

  • $g(x)$ is operator application syntax, not function application. Read Sec.6.4. A function is a set, so we can write $\exists\ f$. An operator $g$ is syntax. Only the value $g(x)$ that results from applying the operator $g$ to the argument $x$ is a set.

    This is the only approach that I have found across the mathematics, computer science, and engineering literature that addresses the question of what "f[x]" means when $x \notin \mathrm{DOMAIN}\ f$. To see what the problem is with traditional notation and education, see this discussion.

  • $f \in [S \rightarrow T]$ says that $f$ is an element of the set of functions with $\mathrm{DOMAIN}$ the set $S$ that take values in the set $T$. Note that "codomain" is not part of the notion of function. Only the range of a particular function makes sense, and the $T$ used to define some specific set of functions.

Using the above notation for the case of functions from the natural numbers (OP), a sequence of real numbers is a function

$f \in [\mathbb{N} \rightarrow \mathbb{R}]$


OLD ANSWER:

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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    Doesn't this cause problems for the order of the elements in the sequence? idea from [this other question](http://math.stackexchange.com/q/323784/190328)2017-01-15
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    The order of the elements in the sequence is the order defined by the linear order between natural numbers. $f[0]$ is the first element, $f[1]$ the second, etc. If you prefer to start counting from 1, then just use $\mathbb{N} \setminus \{0\}$ as the function's domain: $f \in [\mathbb{N} \setminus \{0\} \rightarrow \mathbb{R}]$. For example, in TLA$^{+}$, a tuple is any function indexed over the set $1..n$, for example $\langle 6, 15 \rangle$ is the function $[x \in \{1, 2\} \mapsto \mathrm{IF}\; x = 1\; \mathrm{THEN}\; 6\; \mathrm{ELSE}\; 15$.2017-01-16
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    For the function-notation, there is no problem. I rather meant the set-notation.2017-01-16
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    Well, the notation that looks similar to a set constructor has the $n \in \mathbb{N}$ outside the braces, so it is distinct from $\{x_n : n \in \mathbb{N}\}$. It could be recognized as different from a set constructor, but the difference is small. However, in any case it is informal, so it's meaning is unclear. In contrast, the TLA$^+$ notation is formally defined with axioms, so there isn't any ambiguity there. This is the problem with informal mathematics.2017-01-16