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I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E).   Reach := {u} LastCardinality := 0 while LastCardinality &lt; |Reach| do   LastCardinality := |Reach|   Next := null   foreach i &lt;- Reach do     foreach j &lt;- V do       if(i,j) exists in E then         Next := Next + {j}   Reach := Reach + Next return Reach   So far I have got that the output needs to be Reach = Goal where Goal = {r | r exists V, there is a path from u to r}. I also have three invariants that I am pretty sure will help prove this.   Reach is always a subset of Goal and Next is always a subset of Goal. After an arbitrary K times through the while loop, for every t that exsits in V, if t has a path of length k+1 from u, then t exists in Reach If Reach is a subset of Goal before the while loop then LastCardinality &lt; |Reach|   I know that to finish up the proof, I need to prove termination, the invariants, and correctness but I have no idea how. 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