Prove that the areas of two circles have the same radius as the square of their radii This question is about the provided solution to Question 11d from Chapter 8 of Spivak's Calculus. The relevant background material (i.e. 11a-11c) is as follows:

$(a)$ If $\{a_n\}$ is a sequence of positive terms such that $$a_{n+1}\leq a_n/2$$ then for every $\epsilon>0$ there exists an $n$ with $a_n<\epsilon$.
$(b)$ Let $P$ be a regular polygon inscribed in a circle. Let $P'$ be the regular polygon with twice the sides of $P$. Let $A$ be the area of $P$, $A'$ be the area of $P'$ and $C$ that of the circle. Prove
$$(C-P')\leq \frac 1 2 (C-P)$$
$(c)$ Prove there is a regular inscribed polygon with area as close as that to the circle.

Part d reads as follows:

Using the fact that the areas of two regular polygons with the same number of sides are in the same relation as the squares of their sides, prove the areas of two circles are in the same relation as the squares of their radii.

Here is Spivak's solution (I will include an indexed dagger symbol $\dagger_i$ in places that I will subsequently ask questions about):

Let $r_1$ and $r_2$ be the radii of the two circles $C_1$ and $C_2$, and let $A_i$ be the area of the region bounded by $C_i$. We know that there are numbers $\delta_1, \delta_2 \gt 0$ such that
$$\Big\lvert \frac{A_1}{A_2}-\frac{B_1}{B_2} \Big\rvert \lt \epsilon \quad \dagger_1$$
for any numbers $B_1, B_2$ with $|A_i-B_i| \lt \delta_i \quad \dagger_2$. By part (c), there are numbers $n_i$ such that the area of a regular polygon, with $n_i$ sides, inscribed in $C_i$ differs from $A_i$ by less than $\delta_i$. Let $P_i$ be the area of a regular polygon inscribed in $C_i$ with $\max(n_1,n_2)$ sides. Then
$$\Big\lvert \frac{A_1}{A_2}-\frac{P_1}{P_2} \Big\rvert \lt \epsilon$$
$$\Big\lvert \frac{A_1}{A_2}-\frac{r_1^{\ 2}}{r_2^{\ 2}} \Big\rvert \lt \epsilon$$
Since this is true for each $\epsilon \gt 0$, it follows that $A_1/A_2=r_1^{\ 2}/r_2^{\ 2}$

Referencing $\dagger_1$ and $\dagger_2$, which I view as being directly connected, how is the equation $\Big\lvert \frac{A_1}{A_2}-\frac{B_1}{B_2} \Big\rvert \lt \epsilon$, where $\epsilon$ is arbitrary, derived from the two equations: (1) $|A_1-B_1| \lt \delta_1$ and $|A_2-B_2| \lt \delta_2$. Further, where are the statements $|A_1-B_1| \lt \delta_1$ and $|A_2-B_2| \lt \delta_2$ coming from? (If $B_1$ and $B_2$ are arbitrary numbers, and $\delta_1$ and $\delta_2$ are constructed numbers, I am having difficulties seeing how the $\delta$'s can be related to an arbitrary $\epsilon$)
 A: The map $(0,\infty)\times (0,\infty)\to (0,\infty)$, $(x,y)\mapsto \frac xy$ is continuos. Use the $\epsilon\delta$-definition of continuity (together with a simple lower bound on distances in $(0,\infty)\times (0,\infty)$).

Or calculate directly: If $0<\delta_i<A_i$, then for $|B_i-A_i|<\delta_i$, we have
$$\frac{A_1-\delta_1}{A_2+\delta_2}<\frac{B_1}{B_2}<\frac{A_1+\delta_1}{A_2-\delta_2}.$$
Now show that you can make the $\delta_i$ small enough to ensure
$$ \frac{A_1+\delta_1}{A_2-\delta_2}-\frac{A_1-\delta_1}{A_2+\delta_2}<\epsilon$$
A: Recall the lemmas used in the proofs concerning limits for products, and quotients:

(2) If
$$|x - x_0| < \min\left(1,\frac{\varepsilon}{2(|y_0|+ 1)}\right) \text{ and } |y - y_0| < \frac{\varepsilon}{2(|x_0|+ 1)},$$
then
$$ |xy - x_0y_0| < \varepsilon.$$
(3) If $y_0 \neq 0$ and
$$|y-y_0| < \min\left(\frac{|y_0|}{2}, \frac{\varepsilon|y_0|^2}{2}\right),$$
then $y \neq 0$ and
$$\left|\frac{1}{y} - \frac{1}{y_0}\right| < \varepsilon.$$

Informally, given numbers $x_0$ and $y_0$ with $y_0 \neq 0$, we can make $\frac{x}{y}$ arbitrarily close to $\frac{x_0}{y_0}$ by restricting $x$ and $y$ to be sufficiently close to $x_0$ and $y_0$.
Spivak applies this same argument here. Let $n$ be the number of sides of the regular polygons inscribed in the circles $C_1$ and $C_2$. (The polygons each have $n$ sides. They are both regular $n$-gons). $P_1$ and $P_2$, the areas of these polygons can be made arbitrarily close to the areas of their inscribing circles $A_1$ and $A_2$ by choosing $n$ to be large enough.
Therefore, we can make the ratio $\frac{P_1}{P_2}$ arbitrarily close to $\frac{A_1}{A_2}$.
Spivak then notes that for either polygon, its area is proportional to its incribing circle's radius squared, that is
$$\frac{P_1}{P_2} = \frac{{r_1}^2}{{r_2}^2},$$
where $r_i$ is the radius of circle $C_i$.
As we chose bigger and bigger $n$, $\frac{P_1}{P_2}$ remains constant and equal to $\frac{{r_1}^2}{{r_2}^2}$. And yet, we know by choosing the right $n$ we can get arbitrarily close to $\frac{A_1}{A_2}$. The only way this is possible is if
$$\frac{A_1}{A_2} = \frac{P_1}{P_2} = \frac{{r_1}^2}{{r_2}^2}.$$
That is, the polygon areas get closer and closer to the circle areas, but the ratios remain constant
Finally, note that this is one of those problems that relies a bit on "outside" facts that aren't quite rigorously justified here. That the area of the polygons is proportional to $r^2$ is hopefully not too hard to swallow.
You might find the following geometric argument mildly convincing:
Around the outside of each circle $C_i$ let us construct a square with sides of length $2r_i$ (The circles are themselves inscribed within the squares.)
Now, this argument hinges on the expectation that the polygons, since they each have $n$ sides should "punch out" the same proportion of the areas of their inscribing squares, that is,
$$P_i = k \cdot S_i,$$
where $S_i$ is the area of the square and $k$ is some constant. (To be more precise, we could write $k_n$. $k$ will be different depending on the number of sides, but this doesn't matter for what follows.)
If we accept this, we have
\begin{align}
\frac{P_1}{P_2} & = \frac{k\cdot S_1}{k\cdot S_2} \\ 
& = \frac{S_1}{S_2} \\
& = \frac{2r_1 \cdot 2r_1}{2r_2 \cdot 2r_2} \\
& =\frac{{r_1}^2}{{r_2}^2}.
\end{align}
(IIRC you can avoid this hand wavy "proportional punch out" talk and get the same result on somewhat firmer ground by instead using triangle areas to construct the areas of $n$-gons and so forth.)
This problem provides a cool result, but we're not yet in a position to 100% prove it all with absolute rigor. The concept of "area" hasn't been formally introduced, much less justification for the area formulas for triangles or other polygons.
