Let $P_1,P_2\in\mathbb{R}^3$ and consider all the paths from $P_1$ to $P_2$, I wish to prove that the euclidean distance (that is the length of the line connecting them) is the distanse of the shortest of all paths connecting $P_1$ to $P_2$.
My strategy is to prove that the straight line is a path (trivial) and the for path that is not the straight line connecting $P_1$ to $P_2$ the straight line is shorter, but I am having difficulty with proving the last part.
Can someone please help ?
Edit: for samplicity I edited for the case $n=3$