$\cos(1/x)$ countable on $(0,1)$

Question

Show that the set $S= { x \in (0,1): \cos(\frac{1}{x}) = \pm 1}$ is countable.
From a practice paper. I understand that showing there is a surjection between Natural numbers and a set implies it is countable just unsure on how to do it.
Thanks

One way is to use the Identity Theorem in Complex Analysis for the functions $\cos(\frac{1}{x}) \pm 1$ along with the easily proved fact that every uncountable subset of $(0,1)$ has at least one limit point in $(0,1).$ However, I suspect you want a more direct proof. — Dave L. Renfro, May 07 '14 at 16:33

score 1 · Answer 1 · answered May 07 '14 at 16:05

1

$\cos\frac{1}{x}=\pm 1$ <=> $\frac{1}{x}=n\pi$ <=> $x=\frac{1}{n\pi}$

answered May 07 '14 at 16:05

delta

732

score 0 · Answer 2 · answered May 07 '14 at 16:09

Hint: we can explicitly state the possible values of $x$. The values of $x$ that make $\cos(x)$ equal to $\pm 1$ are the reciprocals of integer multiples of $\pi$. From here, can you see how to create a correspondence between the natural numbers and the values of $x$?

$\cos(1/x)$ countable on $(0,1)$

2 Answers2