Syntax for a sequence in a set X

[imurme8]

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?

 

[JHamm]

I just had a quick flip through Rudin and I only ever saw "a sequence in $\mathbb{R}$".

 

[sachav]

Usually, the set of sequences which elements in a set A is denoted by $A^{\mathbb N}$.
More generally, if A and B are two sets, then the set of functions from A to B is written $B^A$.

 

[micromass]

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?

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 $(x_n)_n$ with each $x_n\in \mathbb{R}$".