# Prove Minkowski's inequality using Cauchy-Schwarz's

## Homework Statement

For u and v in $R^n$ prove Minkowski's inequality that $\|u + v\| \leq \|u\| + \|v\|$ using the Cauchy-Schwarz inequality theorem: $|u \cdot v| \leq \|u\| \|v\|$.

## Homework Equations

Dot product: $u \cdot v = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n$
Norm: $\|u \| = \sqrt {u \cdot u}$
Cauchy-Schwarz inequality: $|u \cdot v| \leq \|u\| \|v\|$

## The Attempt at a Solution

Using def. of norm: $\|u + v\|^2 = \sqrt {(u + v) \cdot (u + v)}^2 = (u + v) \cdot (u + v)$
Expand: $(u + v) \cdot (u + v) = u \cdot u + 2 (u \cdot v) + v \cdot v$
Using the Cauchy-Schwarz inequality: $u \cdot u + 2 (u \cdot v) + v \cdot v \leq \|u \|^2 + 2 \|u \| \|v\| + \| v \|^2 = (\|u \| + \| v \|)^2$

Therefore, $\|u + v\|^2 \leq (\|u \| + \| v \|)^2 \Rightarrow \|u + v\| \leq \|u \| + \| v \|$.

Thank-you