This question rather then wanting help w.r.t (M3) (The Triangle Inequality ) with can be done by using the Cauchy–Schwarz inequality. I would like advice regarding (M1):
$d_E(\mathbf {q,p}) = 0 \iff \mathbf {p =q }\text{ } \forall \mathbf{p,q} \in \Bbb R^n $.
An immediate proof that jump to mind is as follows:
$$d_E = 0 \iff \sqrt{ \sum_{i=1}^n (q_i-p_1)^2 } =0 $$
$$\iff \sum_{i=1}^n (q_i-p_1)^2 =0 $$ and since $q_i, p_i \in \Bbb R \text{ } \forall i \in \{1,...,n\}$
then $(q_i-p_i)^2 =0 $ $\forall q_i \in \mathbf q \land \forall p_i \in \mathbf p$
$\Rightarrow (q_i-p_i) =0 \Rightarrow q_i = p_i \forall i \in \{1,..,n\}$
hence $\mathbf q = \mathbf p$
However I find this proof rather cumbersome. Am I being pandantic? or Is there a more elegant and/or concise way to represent this?