Let $|f\rangle$ be a norm-1 vector in a Hilbert space $H$.
Let $|f_1\rangle,...,|f_n\rangle$ be other norm-1 vectors and let $e_1,...,e_n>0$ with $\sum_{i=1}^n e_i=1$, such that
$$|f\rangle\langle f|=\sum_{i=1}^ne_i|f_i\rangle\langle f_i|.$$
How do I prove that $|f\rangle\langle f|=|f_i\rangle\langle f_i|$ for every $i$?
I truly have no idea where to begin this. Any nudge in the right direction would be greatly appreciated!