 Exponential Chebyshev system

Exponentials with distinct real exponents form a Chebyshev system on every nondegenerate real interval. For an inductive proof, multiply a candidate combination by $e^{-{\lambda }_{0}x}$. If it has $n+1$ zeros, Rolle's theorem gives at least $n$ zeros of its derivative, a combination of the $n$ exponentials with exponents ${\lambda }_i-{\lambda }_0$, $i>0$. The inductive zero bound forces all those coefficients to vanish. The remaining constant also vanishes, a contradiction.