#math #flashcards/math The `triangle inequality` states that given $x,y\in \mathbb{R}$ then %%?%% $ |x+y|\leq{}|x|+|y|. $ The proof is evident by varying positive and negative value combinations of $a$ and $b$. The inequality also extends to $x,y\in \mathbb{R}^{n}$, replacing absolute value with the norm. Suppose we choose definitions of $x,y \in \mathbb{R}^{n}$ such that $x=a-c$ and $y=c-b$ for $a,b,c\in \mathbb{R}^{n}$. Then $ \begin{align} \lVert a-b \rVert & =\lVert x+y \rVert \\ & \leq{}\lVert x \rVert +\lVert y \rVert \\ & =\lVert a-c \rVert +\lVert b-c \rVert . \end{align} $ If $n=1$, this inequality is telling us that the distance from $a$ and $b$ is less than or equal to the summed distance of $a$ to $c$ and $b$ to $c$. If we imagine this with $n=2$, we can imagine $a,b,c$ as points on a plane then the inequality is describing than any side^[The distance between points $a,b$ is $|a-b |$.] is less than or equal to the sum of the other two sides.