$(10$ points$)$ Dijkstra's Algorithm: use the graph below for parts $($b$)$ and $($c$).$

Figure $1$: A Network to test the shortest path algorithm.

$($a$)$ Prove by induction on $m$ that the shortest path algorithm actually does

find the shortest path from a given vertex to the chosen vertex $A.$

$($b$)$ In the graph above, apply the algorithm to find the shortest path from $a$

to $y.$

$($c$)$ Find the shortest path from $d$ to $r.$