Euclidean Metric for Reals in R2

Definition

• With $x,y\in {\mathbb{R}}^2$
• Let $f(x,y)=\sqrt{{\sum }_{i=1}^2(x_i-y_i{)}^2}$

Proof of Metric

1. $(x_i-y_i{)}^2\ge 0,\forall i$
2. Then, ${\sum }_{i=1}^2\ge 0,\forall i$
3. Then, $\sqrt{{\sum }_{i=1}^2(x_i-y_i)}$ is defined $\forall x,y\in {\mathbb{R}}^2$
4. As domain is positive, $f(x,y)>0$