How would you access the following problem:
Show that the set of functions $$ \phi_n : \mathbb{R}_{>0} \rightarrow \mathbb{R}$$$$\phi_n(x) = \frac{1}{n+x}$$for $n \in \mathbb{Z}^{\ge 0}$ is linearly independent.
Since we can not obtain the zero vector (only by dividing a zero) and we can not represent on of the created vectors by combination of others, we know it is linearly independent.
Suppose $$\lambda_1\frac{1}{n_1+x}+\cdots+\lambda_k\frac{1}{n_k+x}=0$$ for all $x$. Each function $\phi_i$ is continuous except at $x=-n_i$, so the sum is discontinuous at $x=-n_i$ except if $\lambda_i=0$.