How to prove that a custom iterative algorithm will determine all shortest paths to a graph node?
You can prove by induction that for every iteration $i$, the array $A$ during this iteration, $A_i$, contains the weights of some paths originating at $t$. More formally, for all $v\in V$, $A_i[v]$ is either $\infty$ or the weight of some path from $t$ to $v$.

Now, it is easy to show that the values of $A_i$ are monotonically non increasing, which combined with the above statement implies that the algorithm terminates. I leave the details to you.