Can it be done in the same way as we prove that the number of points on any two finite line segments are equal - geometrically?
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I don't know how to visualize it geometrically, but the easiest function I can think of for this is $f:(0,1) \to (1,\infty)$ which is $f(x) = 1/x$. – Joshua Ruiter Apr 10 '17 at 13:11
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Are the segment closed? That is, do they include the end points? – ajotatxe Apr 10 '17 at 13:13
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Hint: you can start mapping $(0,1]$ and $[1, \infty)$ with $f(r)=\dfrac 1 r $. – Mauro ALLEGRANZA Apr 10 '17 at 13:13
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You can see the post is-there-a-bijective-map-from $(0,1)$ to $\mathbb R$. – Mauro ALLEGRANZA Apr 10 '17 at 13:45
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1For an open segment (endpoints excluded), bend the segment into a semi-circle and project its points from the center C of the circle onto a line not containing C . If one or both endpoints are included then a continuous bijection is not possible, so use the Cantor-Schroeder-Bernstein theorem. – DanielWainfleet Apr 10 '17 at 21:38