Observations

Useful observations:

• f^*(u) is minimized for u on a shortest (s,t) path, with $f^*(u) = \delta(s,t)$.
• For all u, $f(u) \le f^*(u)$ because of admissibility.
• $f(t) = \delta(s,t)$.
• Consider two consecutive nodes u and v along a shortest path (to anywhere). Claim: $f(u) \le f(v)$. Proof: By monotonicity, $h(u) \le h(v) + w((u,v))$. Because u and v are on a shortest path, $w((u,v)) = \delta(s,v)-\delta(s,u)$. Combining, we get $\delta(s,u) + h(u) \le h(v) + \delta(s,v)$ or $f(u) \le f(v)$.