Hello math lover :
I face some difficulties in solving the problem attached in a file. Can
you help me please.
Thank you in advance
Given any $\displaystyle \epsilon> 0$ If $\displaystyle 0< x< \epsilon$, then $\displaystyle \frac{1}{\epsilon}< \frac{1}{x}< \infty$ which includes an infinite number of periods of sine. That is, sin(1/x) will take on any value between -1 and 1 and is within $\displaystyle \epsilon$ of any point in that set.
$\displaystyle d\left((0,1),\left(x,\sin\left(\tfrac{1}{x}\right) \right)\right)=\sqrt{x^2+\left(1-\sin\left(\frac{1}{x}\right)\right)^2}$. By the Archimedean principle there exists some $\displaystyle n\in\mathbb{N}$ such that $\displaystyle \frac{1}{n}<\varepsilon$ and so $\displaystyle \frac{1}{2\pi n+\frac{\pi}{2}}<\varepsilon$. And so $\displaystyle d\left((0,1),\left(\tfrac{1}{2\pi n},\sin\left(2\pi n\right)\right)\right)=\sqrt{\left(\tfrac{1}{2\pi n}\right)^2+\left(\sin\left(2\pi n+\tfrac{\pi}{2}\right)-1\right)^2}=\frac{1}{2\pi n}<\varepsilon$