Problem 882

Prove: $\sum_{i=1}^{n}\left|z_{i}\right| \leqslant n$.
Example 4 is not difficult, only of medium difficulty level in the national competition, but some students made mistakes by incorrectly using homogeneity. Since the inequality is homogeneous with respect to $a_{1}, a_{2}, \cdots, a_{n}$ and $b_{1}, b_{2}, \cdots, b_{n}$, therefore, without loss of generality, we can assume $\sum_{i=1}^{n} a_{i}=1$ and $\sum_{i=1}^{n} b_{i}=1$, and under these conditions, the problem was solved. In fact, the key in this problem is the case where $\sum_{i=1}^{n} a_{i}=\sum_{i=1}^{n} b_{i}=0$.