Certain property of finite field $F_p$ Let $F_p$ be a finite field of order $p$ where $p$ is a prime number. Let $\{ \alpha_1, \ldots, \alpha_{p-1} \}$ be a multi-set with $\alpha_i \in F_p$ and each $\alpha_i$ non-zero. 
I want to show that $$\sum_{i\in K} \alpha_i = -1$$ for some subset $K \subseteq \{1,\ldots, p-1\}$. 
I am stuck at this. Any hint(s) would be appreciated. 

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*Since any element $\alpha \in  \mathbf{F}_p^*$ generates $(\mathbf{F}_p,+)$, the only subset $S \subset \mathbf{F}_p$ such that $S= S \cup (\alpha+S) \bmod p$ is $\mathbf{F}_p$.

*If all the $\alpha_i$ are the same element $\alpha$, take $b \equiv -\alpha^{-1} \bmod p$ so that $\sum_{i=1}^b \alpha_i = -1$.

*Otherwise wlog. $\alpha_1 \ne \alpha_2$ so that $S_2 = \{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}$ contains $3$ elements, and $S_{i+1} = S_i \cup ( \alpha_{i+1}+S_i) $ contains at least $i+2$ elements, therefore $S_{p-1}$ contains $-1$.
