Given $ \alpha\_1 > \alpha\_2 > \cdots > \alpha\_n \geq 0$, $f\_1(x)= e^{-\alpha\_1x},..., f\_n(x)=e^{-\alpha\_nx}$, prove that they are linearly independent . Hint: don’t forget the limit $ x \rightarrow \infty $.

I have already seen proofs about similar questions. What you have to do is to set the combination of those functions (or polynomials) equal to 0 and show that all coefficients have to be 0. But here, the coefficients are in the exponent, they can’t be equal to each other, and the negative exponent disturbs me a bit.

Thanks for your help.