A set $A$ contains infinitely many terms of a real sequence $\{x_n\}_{n=1}^{\infty}$

Question

What does this statement mean mathematically ?

"A real sequence $\left\{x_n\right\}_{n=1}^{\infty}$ is contained in set $A$ for infinitely many $n \in \mathbb{N}$."

This is my mathematical statement

"A real sequence $\left\{x_n\right\}_{n=1}^{\infty}$ is said to be contained in set $A$ for infinitely many $n \in \mathbb{N}$, if there exist a one to one correspondence from the set $\left\{n \in \mathbb{N} \textrm{ } | \textrm{ } x_n \in A \right\} $ to $\mathbb{N}$."

Is my statement a correct mathematical representation of the statement "A real sequence $\left\{x_n\right\}_{n=1}^{\infty}$ is contained in set $A$ for infinitely many $n \in \mathbb{N}$." ?

If not, then what would be the correct mathematical definition for the former statement ?

Answer 1

Your statement is correct now. But to a mathematician, perhaps a more natural formalisation would be:

For all $n\in\mathbb N$ there exists $m\in\mathbb N$ with $m\ge n$ and $x_m\in A$

This is about individual elements rather than correspondences between sets, which makes it easier to work with.