I have been struggling with understanding the two properties that arise if we define a metric in terms of the norm $d(x,y) = ||x - y||$.
It is said that such a metric then has the following properties: $$d(x + z, y + z) = d(x,y)$$ translation invariance, and $$d(ax, ay) = |a| d(x,y)$$ homogeneity.
Now by the answers on this site I have seen that these properties are required to induce a norm from that metric (the converse) and that other metrics not induced by a norm might not have these two properties.
I am having problems seeing how these properties appear when a metric is defined in terms of the norm. I suppose that the homogeneity property appears like this: $$d(ax,ay) = \lVert a x-a y\rVert=\bigl\lVert a(x-y)\big\rVert=\lvert a\rvert\lVert x-y\rVert = |a|d(x,y).$$ Correct me if I am wrong, but this one is directly related to the homogeneity property of the norm so it makes sense. But how does this translation invariance appear?
\begin{align*} d(x+z, y+z) & = \| (x + z) - (y + z) \| \\ & = \| x - y \| \\ & = d(x, y) \end{align*}