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Syntax for a sequence in a set X

  1. Dec 30, 2011 #1
    Syntax for a sequence "in" a set X

    It is well-known shorthand to say that a sequence of real numbers x_n is a sequence "in R". (Of course we do not mean that the function x_n is an element of R). In such a case, is it permissible to replace the word "in" by the element symbol, or is this not accepted usage?

    E.g. "Given a sequence {x_n} \in \mathbb{R}..." Is this OK?
  2. jcsd
  3. Dec 30, 2011 #2
    Re: Syntax for a sequence "in" a set X

    I just had a quick flip through Rudin and I only ever saw "a sequence in [itex]\mathbb{R}[/itex]".
  4. Dec 30, 2011 #3
    Re: Syntax for a sequence "in" a set X

    Usually, the set of sequences which elements in a set A is denoted by [itex]A^{\mathbb N}[/itex].
    More generally, if A and B are two sets, then the set of functions from A to B is written [itex]B^A[/itex].
  5. Dec 30, 2011 #4


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    Re: Syntax for a sequence "in" a set X

    No, this is not permissible. This would be a serious abuse of notation and it would likely cause trouble.

    One should say "Given a sequence [itex](x_n)_n[/itex] with each [itex]x_n\in \mathbb{R}[/itex]".
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