Complex Analysis: Theory and Applications · Problem 1.3.23

This is the triangle inequality for sums — a foundational property showing that the magnitude of a sum never exceeds the sum of magnitudes. The key insight is that induction lets you apply the two-term triangle inequality $|z + w| \leq |z| + |w|$ repeatedly: once you know the inequality holds for $n$ terms, you can treat their sum as a single complex number and add one more term, letting the base case do the heavy lifting. This pattern of "assume it works for $n$, then show it works for $n+1$" is exactly what induction is built for.

This is the triangle inequality for sums — a foundational property showing that the magnitude of a sum never exceeds the sum of magnitudes. The key insight is that induction lets you apply the two-term triangle inequality $|z + w| \leq |z| + |w|$ repeatedly: once you know the inequality holds for $n$ terms, you can treat their sum as a single complex number and add one more term, letting the base case do the heavy lifting. This pattern of "assume it works for $n$, then show it works for $n+1$" is exactly what induction is built for.