I got a weighted, connected and directed graph $G$. There is a vector called the distance vector $Dv \in \mathbb{N}^n$ in which $Dv_i$ is the shortest distance from $1$ to $i$. All edge weights are positive integers. I have to show that every distance vector $Dv$ satisfies:$Dv_1 = 0$.For all $j \neq 1$ there exists $i$ such that $Dv_j = Dv_i + w(i,j)$.For all $i,j$ it holds that $Dv_j \leq Dv_i + w(i,j)$.I think that 1 is trivial: from 1 to 1 you have no distance. But the rest? Can you give me an idea how to prove 2 and 3? 
Distance vector in a weighted graph
algorithms;graphs