The triangle inequality for vectors states that ∥u+v∥\|\mathbf{u} + \mathbf{v}\|∥u+v∥:
=∥u∥+∥v∥= \|\mathbf{u}\| + \|\mathbf{v}\|=∥u∥+∥v∥ always
≥∥u∥+∥v∥\geq \|\mathbf{u}\| + \|\mathbf{v}\|≥∥u∥+∥v∥
≤∥u∥+∥v∥\leq \|\mathbf{u}\| + \|\mathbf{v}\|≤∥u∥+∥v∥
=∣∥u∥−∥v∥∣= |\|\mathbf{u}\| - \|\mathbf{v}\||=∣∥u∥−∥v∥∣