Show the independence of $\{f_n(x)=\frac1{n+x}\space|\space n \in\mathbb N\}$ I'm currently trying to solve the following problem:
Show that the set of vectors $\{f_n(x)=\frac1{n+x}\space|\space n\in\mathbb N\}$ over the vector space of the functions from $\mathbb{R_{>0}}$ to $\mathbb R$ is linear independent.
I have managed to show it, but I used the geometric expansion and the determinant of the Vandermonde matrix, which is far to advanced for our current level. Is there a simpler way to approach it?

Suppose $$\sum_1^N{a_n\over n+x}$$ is identically zero. Clearing fractions, $$p(x)=\sum a_nq_n(x)$$ is identically zero, where $q_n(x)=((1+x)\cdots(N+x))/(n+x)$. But if $a_n\ne0$, then $p(-n)\ne0$. Hence, $p(x)$ is identically zero if and only if $a_n=0$ for all $n$. 
