I was doing the OMU test (https://www.olimpiada.ime.unicamp.br/), witch is already done (just to be clear), and came across the following question:
Consider $C$ the circle with center in $(0, 0)$ and radius r = 1, that is: $$C = \{(x, y) : x^2 + y^2 \leq 1\}$$ Given any natural number $n> 0$, show that there is a function $F:[0, 1]\longrightarrow C$ so that:
The image set of $F$, denoted by $Im(F)$ is a regular polygon with at least n sides and (I)
$|Area(Im(f)) - \pi| < \frac{1}{n}$ is true (II)
Here is what I made:
If $Im(F)$ describes a poligon with p sides inscribed in $C$ and $p\longrightarrow \infty$, (II) is true, because the polygon area tends to be equal to circle area. So: \begin{gather} |\lim_{p\rightarrow \infty} Area(Im(f))-\pi| < \frac{1}{n} \\ |(\pi*1)-\pi| < \frac{1}{n}\\ 0 < \frac{1}{n}\quad (true) \end{gather} And (I) is true because: $$[0, 1]\sim\mathbb{R}^2$$ and $$\mathbb{R}^2\supseteq C \supseteq Im(f)$$ So there is a injection function from $[0, 1]$ to $Im(F)$ no matter $Im(F)$.
Have I made every thing right? And this was from the high school level so I supose there is another simplier way to solve, am I right? Could anyone help me?