# Zadanie 8


$||\vec{U}|| \cdot ||\vec{V}|| = \sum_{i=1}^{n} |u_i| \cdot \sum_{i=1}^{n} |v_i|$

$\vec{U} \cdot \vec{V} = \sum_{i=1}^n u_i \cdot v_i \leq \sum_{i=1}^n |u_i| \cdot |v_i|$

$\sum_{i=1}^{n} |u_i| \cdot \sum_{i=1}^{n} |v_i| = (|u_1| + |u_2| + ... + |u_n|) \cdot (|v_1| + |v_2| + ... + |v_n|) = |u_1| |v_1| + |u_2| |v_2| + ... + |u_n| |v_n| + r =$

$= \sum_{i=1}^n |u_i| \cdot |v_i| + r$

Ponieważ $r \geq 0$ to:

$\sum_{i=1}^n |u_i| \cdot |v_i| \leq \sum_{i=1}^n |u_i| \cdot |v_i| + r$

$\sum_{i=1}^n |u_i| \cdot |v_i| \leq \sum_{i=1}^{n} |u_i| \cdot \sum_{i=1}^{n} |v_i|$

$\vec{U} \cdot \vec{V} \leq ||\vec{U}|| \cdot ||\vec{V}||$