I am new to functional Analysis and the way convergence of a sequence is defined confuses me . I am reading the book by Kreyzig where he says : A sequence ${(x_n)}$ in a metric space $X =(X,d)$ is said to converge if there is an $x \in X$ such that $$ \lim_{n \to \inf} d(x_{n},x) = 0$$
x is called the limit of $x_{n}$ and we write : $$ \lim_{n \to \inf} x_{n} = x$$
Further , the author says that the metric d yields the sequence of real numbers : $$a_{n} = d(x_{n},x)$$ whose convergence defines that of $x_{n}$.
My Questions are
1.) When author says $x \in X$ , does it mean $x$ is a sequence ? Intuitively $x$ should be a number as it is the limit of a sequence . BUT
2.) If $x$ is a number then how can we calculate the distance which is defined on sequences ,given that we are in a sequence space.
3.)How can the metric $d$ yield a sequence $a_{n}$ and not a number ?
1.) x is an element of X. If X is $\mathbb{R}$ then yes, x is just one real number.
To see why we care about $x \in \mathbb{R}$, consider if $X = (0,1)$. A sequence like $\frac{1}{n}$ will not (and should intuitively) not converge because $0$ is not in the set $X$.
2/3.) Kreyzig is just defining a new sequence $\{a_n\}$. Every metric space will have some metric d. It might be helpful to just think about this as 'distance'. How he is defining $\{a_n\}$ is by taking the distance from every point in the sequence $\{x_n\}$ and subtracting the point to which the sequence converges. See that this sequence should then converge to $0$.
Note when he writes $a_n = d(x_n, x)$ he is defining a term of the sequence. Hence $\{a_n\} = \{a_1, a_2,... \}$.