The modulus of a sum

Expansion.
$$|z_1+\cdots+z_n|^2=\sum_i|z_i|^2+2\operatorname{Re}\sum_{i<j}z_i\bar z_j.$$

Bound. $\operatorname{Re}(z_i\bar z_j)\le|z_i\bar z_j|=|z_i||z_j|$, so
$$|z_1+\cdots+z_n|^2\le\sum|z_i|^2+2\sum_{i<j}|z_i||z_j|=\left(\sum|z_i|\right)^2.$$
This gives $|z_1+\cdots+z_n|\le\sum|z_i|$, the triangle inequality.

Sharper bound. By Cauchy-Schwarz applied to the vectors $(1,\ldots,1)$ and $(|z_1|,\ldots,|z_n|)$:
$$\left(\sum|z_i|\right)^2\le n\sum|z_i|^2.$$
Combining: $|z_1+\cdots+z_n|^2\le n\sum|z_i|^2$.

Conclusion. The inequality holds.