Let $a(s)$ be a regular curve that is parametrized by arclength. Prove that, if the curvature $k(s)$ is a strictly monotonic function, then $a(s)$ has no self intersections. Suggestions:

a) [will be ommited since I managed to prove it using this first suggestion so there's no point putting it here]

b) Consider the curve $b(s) = a(s) + \frac{n(s)}{k(s)}$, with $k'(s) > 0$. Verify that for all $s > s_0$, the following are true, and then conclude that $a(s)$ has no self intersections:

$|b(s) - b(s_0)| <\int_{s_0}^{s} |b'(s)| ds = \frac{1}{k(s_0)} - \frac{1}{k(s)}$

$|b(s) - b(s_0)| < \frac{1}{k(s_0)}$

I managed to compute the first integral, but I'm lost on how to prove the other two inequalities. I would appreciate any ideas. I know there are other ways to prove the statement, but I want to do it specifically using the second suggestion. Also, I don't know where the contradiction in b) is, I'm not seeing how both being true is not possible.


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