At all times, for all v, Dist(v) is the length of shortest path from s to v that only goes through vertices in V-U
Induction Step: Suppose true for first k steps. The SP to the (k+1)st closest vertex, say w, can go through only vertices in V-U, otherwise, there would be a closer vertex. Therefore, selecting the min => add the k+1st.
Say w is added.
New Dist value for a vertex x is min of old Dist value and Dist(w) + c(w,x)