Is $Y := \{(x, \cos\frac{1}{x}) :\, 0 < x \leq 1\}$ compact?

Question

Is $Y := \{(x, \cos\frac{1}{x}) :\, 0 < x \leq 1\}$ compact?

I have a lot of theorems about compactness. But I am not sure which one to use and how to solve it and I couldn't find similar example on the internet. Can anybody help me?

Answer 1

Hint. Prove that $Y$ is not closed by exhibiting a sequence $(x_n)$ of $(0,1]$ such that $(x_n,\cos(\frac{1}{x_n}))\to (0,1)\notin Y.$

Answer 2

Hint: what happens as $x \to 0+$?