In Michael Spivak's Calculus to explain limits, he uses such an example:
We're given a function $f(x) = \left\{0, \quad \quad \quad \quad x \,\,\, \text{irrational} \\ \quad\quad\quad\quad\frac{1}{q}, \quad \quad \quad x = \frac{p}{q} \,\,\,\text{in lowest terms},\,\, 0<x<1 \right\}$
And we are told that to prove that for any number $a$ with $0<a<1$ the function approaches a at $a$, we consider any number $\epsilon > 0$ and any natural number $n$ large enough, that $\frac{1}{n}\leq \epsilon$.
And here is the part I don't get:
that $1/n\le\varepsilon$. Notice that the only numbers $x$ for which $|f(x)-0|< \varepsilon$ could be false are: $$\frac{1}{2};\frac{1}{3};\frac{2}{3};\frac{1}{4};\frac{3}{4};\frac{1}{5};\frac{2}{5};\frac{3}{5};\frac{4}{5};\ldots;\frac{1}{n},\ldots,\frac{n-1}{n}.$$ (If $a$ is rational, then $a$ might be one of these numbers.) However many of these numbers there may be, there are, at any rate, only finitely many.
What does he mean by only finitely many?
Can't I just go on $\frac{1}{2}, \frac{1}{2}, \frac{1}{4}, \frac{1}{5}, \frac{1}{500}, \frac{1}{5000000000}, \frac{1}{500000000^{5000000000}}$ and so on?
Also then he says that if the distance between $x$ and $a$ is less than the distance between the difference between the closest rational number betewen $0$ and $1$, and $a$, then $x$ is not one of the rational numbers between $0$ and $1$ or in other words:
Set $A$ contains all rational numbers between $0$ and $1$
From all members of $A$, $\frac{p}{q}$ is closest to $a$
$|\frac{p}{q} - a| = \delta$
if $|x - a| < \delta$, then $x$ does not belong in $A$
I don't get it - why doesn't $x$ belong in $A$?It all seems like random assumptions coming out of nowhere, I'm not sure I'm following.Could someone clarify this here?
With regards to the first part he is saying that the only numbers for which that function can yield a number that is greater than $\epsilon$ are those whose denominator is $n$ or less, hence the numbers he listed. The number $n$ is fixed at the beginning if that's what's causing confusion. For a specifically chosen $n$ the numbers in the sequence you present eventually become less than $\frac{1}{n}$