 Solution

The unique root-to-$u$ path in $T$ is the reversal of its predecessor chain. Every edge satisfies $w(e_z)=d(z)-d(p(z))$. Summing along that chain telescopes tosince $d(v)=0$ for positive weights. Thus the tree path is a shortest $vu$-path in $G$ for every $u$, and $T$ is a shortest-path tree rooted at $v$.