Understanding proof of uniform continuity on an interval

Question

I do not quite understand the method of taking a sequence from some interval. $$ is supposed to be an interval, so how from $$ do we form a sequence? And then how do we 'get' two sequences $x_n$ and $y_n$ from points in $R$? And how from thay do we know that $|x_n - y_n| < ε $ . How do they deduce that $x_n$ is bounded? How do I interpret $f(x_{n})$ or $f(x_{n_k})$ ? Clearly I am missing some prerequisite knowledge with which I should understand this proof. If anyone could explain me the questions I asked would be nice.

Answer 1

The first occurrence of $\delta$ is "$\forall \delta > 0$", that is, $\delta$ is a positive real number (not an interval). And note that the statement in (3.5) is true for all positive real numbers $\delta$, and for each choice of $\delta$, there are some $x, y \in [a, b]$ satisfying the stated properties. So, for each $n$, choosing $\delta_n = 1/n$, we get some $x_n, y_n \in [a, b]$ with the properties stated in (3.5). This is how the sequences $(x_n)_{n \geq 1}$ and $(y_n)_{n \geq 1}$ with the desired properties are constructed.

The notation $f(x_n)$ and $f(x_{n_k})$ means exactly what it looks like: it's the output of the function $f$ given the inputs $x_n$ or $x_{n_k}$ (which are in the domain of $f$).

Answer 2

$\delta$ is an arbitrary positive number. By letting $\delta = 1/n$ for each $n \in \mathbb{N}$, we get $x_n,y_n\in [a,b]$ such that whenever $|x_n-y_n|<\delta_n$, we have $|f(x_n)-f(y_n)|<\epsilon$. This gives us the described sequences $(x_n)$ and $(y_n)$.

$(x_n)$ is bounded because each $x_n$ belongs to $[a,b]$.

$f(x_n)$ is the result of plugging $x_n$ into $f$.