The proof of Extreme Value Theorem (part 1) states:
If $f(x)$ is continuous on $[a,b]$, then it is bounded on $[a,b]$.
I'm confused with this part of the proof:
Suppose $f(x)$ is not bounded; then, for every natural number $n$, there exists an $x_n \in [a,b]$, such that $f(x_n) > n$
My question is, how we can give some order to rational numbers in this interval $[a,b]$? We cannot count rational numbers.
Is this enumeration done with respect to the magnitude of numbers from $[a,b]$, or is it just random enumeration?
Can we just start the proof like this?
Suppose $f(x)$ is not bounded; then there is no natural number that is greater than or equal to $f(x)$ where $x \in [a,b]$?