Prove some properties of a given distance function

Let $A, B$ be points in $\mathbb{R}^n$ and define the distance between $A$ and $B$ by

$d(A,B) = \lVert A - B \rVert.$

Prove the following:

Proof. We compute,
$\begin{align*} d(A,B) &= \lVert A - B \rVert \\ &= \left( \sum_{k=1}^n (a_k - b_k)^2 \right)^{\frac{1}{2}} \\ &= \left( \sum_{k=1}^n (b_k - a_k)^2 \right)^{\frac{1}{2}} \\ &= \lVert B - A \rVert \\ &= d(B,A). \qquad \blacksquare \end{align*}$
Proof. Assume $d(A,B) = 0$ , then
$\begin{align*} && d(A,B) &= 0 \\ \implies && \left( \sum_{k=1}^n (a_k - b_k)^2 \right)^{\frac{1}{2}} &= 0 \\ \implies && a_k - b_k &= 0 & \text{for all } 1 \leq k \leq n \\ \implies && A &= B. \end{align*}$

Conversely, assume $A = B$ . Then $a_k - b_k = 0$ for all $1 \leq k \leq n$ . Therefore,

$d(A,B) = \left( \sum_{k=1}^n (a_k - b_k)^2 \right)^{\frac{1}{2}} = 0. \qquad \blacksquare$
Proof. We compute
$\begin{align*} d(A,B) &= \lVert A - B \rVert \\ &= \lVert A-C + C - B \rVert \\ &= \lVert (A-C) + (C-B) \rVert \\ &\leq \lVert A- C \rVert + \lVert C-B \rVert &(\text{triangle inequality}) \\ &= \lVert A - C \rVert + \lVert B - C \rVert &(\text{part (b)}) \\ &= d(A,C) + d(B,C). \qquad \blacksquare \end{align*}$