I'm looking for a proof using elementary arguments of the following fact :
Let $a_1<a_2<\cdots<a_n$ and $\lambda_1,\ldots, \lambda_n$ be real numbers. One consider the function defined over $\mathbb R$ by : $$f: x\mapsto \sum_{k=1}^n\lambda_k e^{a_kx}.$$ If $f$ vanishes at $n$ distinct points then $\lambda_1=\cdots = \lambda_n=0$.
I know that the functions $x\mapsto e^{a_k x}$ with $k=1,\ldots,n$ are linearly independant but the condition isn't really the same here. This could be related to Vandermonde matrix but the calculus are ugly. So I'm trying to find an other way to proceed.
Any idea ?
