As a part of my question I am attempting to prove that $f \in C^\infty( \left] 0,\infty \right[)$ whereby $f(x)=e^{-\frac{1}{x}}, x>0$.
How do I go about doing this.
My idea ist to look at $g(x)=\exp(x)$ and $h(x)=-\frac{1}{x}$ and prove that they're both $\in C^\infty(\left]0,\infty\right[)$ and using the chain rule show that $f$ is infinitely differentiable.
However, how would I even prove that? Surely, finding the nth derivative using induction is by no means proving differentiability?
Any help is greatly appreciated.