Let $\alpha : [0, + \infty) \rightarrow \Bbb{R}^2$ be a map defined as $\alpha(0) = (0,0)$ and $\forall t > 0 : \alpha (t) = (e^{-1/t} \cos (1/t) , e^{-1/t} \sin (1/t)) $ . Prove that $\alpha$ is $C^\infty$ on all $[0, +\infty) $ .
I am posting this excercise from my Differential geometry book since I think that it is very easy but I am very stuck and I do not see it clearly.
My problem is that, since obviously $\alpha$ is $C^\infty$ on all $(0, +\infty) $ because on that interval it is a composition and product of $C^\infty$ maps, I have tried to search a pattern on $\alpha^{(k)}$ (the $k$ derivative of $\alpha$) in order to justify its continuity on $0$ by applying the definition of derivative on a specific point (I mean $\alpha^{(k+1)} (0) = \lim_{h \rightarrow 0} \frac{\alpha^{(k)}(h) - \alpha^{(k)}(0)}{h}$ ) , but I do not see any pattern related to $k$ when I start to derivate, so any possible help or advice would be very appreciated.