Proving an inequation about the distance in triangles

Question

I could need some help with this, Write down a prove, that shows that the following inequation is true for all a, b, c ∈ R^n d(a, b) ≤ d(a,c) + d(c,b)

I'm thankful for every input

Answer 1

Assuming $d$ is defined as $d(a,b)=\|a-b\|$, then you can prove this using the triangle inequality of the norm in $\Bbb{R}^n$ : $$\begin{align} d(a,b)=\|a-b\|&=\|(a-c)+(c-b)\| \\ &\leq \|a-c\|+\|c-b\| \\ &=d(a,c)+d(c,b). \end{align}$$