triangle_inequality

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Definition

For all $a, b \in \mathbb{R}$,

$$ |a + b| \leq |a| + |b| $$

More generally, for any finite collection of real numbers,

$$ \left| \sum_{i=1}^{n} a_i \right| \leq \sum_{i=1}^{n} |a_i| $$

Proof

We have $-|a| \leq a \leq |a|$ and $-|b| \leq b \leq |b|$.

Adding: $-(|a| + |b|) \leq a + b \leq |a| + |b|$.

This means $|a + b| \leq |a| + |b|$.

The general case follows by induction.