In Chapter 1 of Pugh's Real Mathematical Analysis, Pugh gives the following picture:
I'm aware of other proofs to this like this one: bijection from (a,b) to R but I'm interested in understanding how I should be interpreting this image. From top to bottom, i see the unit circle, the (-1, 1) line, the (a,b) line but I'm not sure what the lines connecting the intervals are meant to show. Would something like this be sufficient for a proof or is Pugh just trying to give intuition?
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I think the picture certainly helps with intuition as well as providing a sort of visual proof. You can see that there is a bijective correspondence. I don't guess I have alot to add. Just look and see how the mappings work. As to an analytic description of the functions, not clear what you would get. But that isn't really important from the point of view of the claim.
The diagram shows two bijections. The first, from $(a,b)$ onto $(-1,1)$, is a straightforward projection; call it $p$. Let $P$ be the point at the bottom where a bunch of lines converge. Then for each $x\in(a,b)$ we find $p(x)\in(-1,1)$ by drawing a line from $P$ through $x$ to the segment $(-1,1)$: the point where it hits the segment is $p(x)$.
The second bijection is from $(-1,1)$ to $\Bbb R$ and is a little more complicated; call it $g$. It uses the semicircle of radius $1$ centred at $C=\langle 0,1\rangle$ above the centre of the segment $(-1,1)$. If $x\in(-1,1)$, we find $g(x)$ as follows. First draw a vertical line up from $x$ until it meets the semicircle, say at a point $X$. Then draw a line from $C$ through $X$ to the straight line containing the segment $(-1,1)$; the point of intersection with that line is $g(x)$. As $x$ moves towards either end of the interval $(-1,1)$, the line from $C$ through $X$ gets closer and closer to horizontal, and $g(x)$ moves further off towards $-\infty$ or $+\infty$.
The function $f$ in the diagram is $g\circ p$, the composition of $p$ with $g$; it’s a bijection from $(a,b)$ to $\Bbb R$.