I'm reviewing my old algorithms notes and have come across this proof. It was from an assignment I had and I got it correct, but I feel that the proof certainly lacks.

The question is to prove that the distance values taken from the priority queue in Dijkstra's algorithm is a non-decreasing sequence.

My proof goes as follows:

Proof by contradiction. Fist, assume that we pull a vertex from Q with d-value 'i'. Next time, we pull a vertex with d-value 'j'. When we pulled i, we have finalised our d-value and computed the shortest-path from the start vertex, s, to i. Since we have positive edge weights, it is impossible for our d-values to shrink as we add vertices to our path. If after pulling i from Q, we pull j with a smaller d-value, we may not have a shortest path to i, since we may be able to reach i through j. However, we have already computed the shortest path to i. We did not check a possible path. We no longer have a guaranteed path. Contradiction.