Prove Minkowski's inequality using Cauchy-Schwarz's

[logan3]

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

 

[jbunniii]

Looks fine to me.