Why Does Spivak Exclude Specific Rational Points in His Limit Proof?

[Bashyboy]

Homework Statement

Consider the function
$$
f(x) := \begin{cases}
0, & if~ x \in (0,1) - \mathbb{Q} \\
\frac{1}{q}, & if~ x = \frac{p}{q} \in (0,1) \cap \mathbb{Q} \mbox{ in lowest terms. } \\
\end{cases}
$$

In his Calculus book, Spivak shows us that $\lim_{x \rightarrow a} f(x) = 0$ for all $a \in (0,1)$. I am having trouble with a few of his remarks

Homework Equations

The Attempt at a Solution

First, he "let $n$ be a natural number so large that $\frac{1}{n} \le \epsilon$." This is fine, I suppose; this is just the Archimedean property, although I have always taken $\frac{1}{n}$ strictly less than $\epsilon$. He then goes on to say "that the only numbers $x$ for which $|f(x)-0| < \epsilon$ could be false are:

$$\frac{1}{n},...,\frac{n-1}{n}$$"

I find this rather perplexing...Why wouldn't $x = \frac{1}{n}$ satisfy $|f(x)| < \epsilon$, for example? Spivak's aim seems to be choosing a $\delta$ such that any $x$ in $(a-\delta, a+\delta)$ is not anyone of the numbers in the above list but is smaller, so that $|f(x)| < \epsilon$. But, as I ask, why wouldn't $x = \frac{1}{n}$ work?

PS I can't figure out why there are horizontal lines through my text.
Mod note: Fixed this. For some reason the overstruck text was surrounded by overstrike BB code tags -- [ s] and [ /s]. I removed them.

 

[Ray Vickson]

Homework Statement

Consider the function
$$
f(x) := \begin{cases}
0, & if~ x \in (0,1) - \mathbb{Q} \\
\frac{1}{q}, & if~ x = \frac{p}{q} \in (0,1) \cap \mathbb{Q} \mbox{ in lowest terms. } \\
\end{cases}
$$

In his Calculus book, Spivak shows us that $\lim_{x \rightarrow a} f(x) = 0$ for all $a \in (0,1)$. I am having trouble with a few of his remarks

Homework Equations

The Attempt at a Solution

First, he "let $n$ be a natural number so large that $\frac{1}{n} \le \epsilon$." This is fine, I suppose; this is just the Archimedean property, although I have always taken $\frac{1}{n}$ strictly less than $\epsilon$. He then goes on to say "that the only numbers $x$ for which $|f(x)-0| < \epsilon$ could be false are:

$$\frac{1}{n},...,\frac{n-1}{n}$$"

I find this rather perplexing...Why wouldn't $x = \frac{1}{n}$ satisfy $|f(x)| < \epsilon$, for example? Spivak's aim seems to be choosing a $\delta$ such that any $x$ in $(a-\delta, a+\delta)$ is not anyone of the numbers in the above list but is smaller, so that $|f(x)| < \epsilon$. But, as I ask, why wouldn't $x = \frac{1}{n}$ work?

PS I can't figure out why there are horizontal lines through my text.

I had such horizontal lines in one of my posts about two weeks ago. The only cure was to log off and re-boot my computer. It worked for me, maybe not for you.

 

[Mark44]

I had such horizontal lines in one of my posts about two weeks ago. The only cure was to log off and re-boot my computer. It worked for me, maybe not for you.

See my added note in the first post of this thread.