A subset $A$ of a space $X$ is called dense if every point $x$ in $X$ either belongs to $A$ or is a limit point of $A$ ; that is, the closure of $A$ constitutes the whole set $X$.
That the definition of a dense set, what about sequences? what is the meaning of dense sequence?
3 Answers
A sequence can be viewed as a set by forgetting the order of the elements. The sequence $(x_n)_{n\in\Bbb N}$ is dense if the set $\{x_n:n\in\Bbb N\}$ is dense.
A sequence is dense if the set whose elements are the sequence elements, is dense according to your definition.
A sequence is not dense. It’s a common abuse of terminology that identifies the sequence, which is a function from the natural numbers to the real numbers (usually), with its image set, which is a subset of the codomain of the sequence.
A sequence $(x_n)_{n}$ is a function from the natural numbers to a set $X$ and, formally, can be written as $(x_n)_{n} \in X^{\mathbb{N}}$. The image of any function is a subset of the codomain, and so we write $\{x_n : n \in\mathbb{N}\}$. Hence, to be precise, we should say that the image set of the sequence is dense in $X$, that is $\overline{\{x_n : n \in\mathbb{N}\}}=X$
The same Rudin, Principles of Mathematical Analysis, definition 2.7: "By a sequence, we mean a function $f$ defined on the set $J$ of all positive integers".
