I'm reading the classical Do Carmo's differential geometry book and I'm having trouble to prove the item b of the question 7 from section 1.5 (page 23):
My attempt
My attempt was to fix a normal line $l_1:\alpha(t_1)+\lambda n(t_1)$ and take a function $f(t)$ which gives me the intersection point of the normal line at $\alpha(t)$ and $l_1$ for each $t$. Afterwards, I took the limit $t\to t_1$.
This approach didn't give me anything, just a cumbersome $0/0$ limit.
Any hint? maybe a simpler approach?
This is not a very rigorous proof but...
You proved in part a, that normal to $\alpha$ at $t$ is tangent at $t$ to evolute of $\alpha$ at $t$
So, normal to $\alpha$ at $t_1$ is tangent at $t_1$ to evolute of $\alpha$ ,$\quad$ and normal to $\alpha$ at $t_2$ is tangent at $t_2$ to evolute of $\alpha$, where $t_1$ and $t_2$ are very close points on the evolute.
So, as $t_1$ approaches $t_2$, tangents at very close points on a curve will converge to a point on the curve (the evolute here).