Euclidean Metric on Real Vector Space is Metric/Proof 2

Proof
Consider the Euclidean space $M = \struct {\R^n, d_2}$ where $d_2$ is the distance function given by:
 * $\ds \map {d_2} {x, y} = \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{\frac 1 2}$

where $x = \tuple {x_1, x_2, \ldots, x_n}$ and $y = \tuple {y_1, y_2, \ldots, y_n}$.

Proof of $\text M 1$
So axiom $\text M 1$ holds for $d_2$.

Proof of $\text M 2$
It is required to be shown:
 * $\map {d_2} {x, y} + \map {d_2} {y, z} \ge \map {d_2} {x, z}$

for all $x, y, z \in \R^n$.

Let:
 * $(1): \quad z = \tuple {z_1, z_2, \ldots, z_n}$
 * $(2): \quad$ all summations be over $i = 1, 2, \ldots, n$
 * $(3): \quad x_i - y_i = r_i$
 * $(4): \quad y_i - z_i = s_i$.

Thus we need to show that:
 * $\ds \paren {\sum \paren {x_i - y_i}^2}^{\frac 1 2} + \paren {\sum \paren {y_i - z_i}^2}^{\frac 1 2} \ge \paren {\sum \paren {x_i - z_i}^2}^{\frac 1 2}$

We have:

So axiom $\text M 2$ holds for $d_2$.

Proof of $\text M 3$
So axiom $\text M 3$ holds for $d_2$.

Proof of $\text M 4$
So axiom $\text M 4$ holds for $d_2$.