There are 2 sets , $A=\{a_1,a_2,a_3..... ,a_n\} $ and $B=\{b_1,b_2,b_3.. b_n\}$ . All elements of these sets are real numbers. Each point in set $A$ need to be connected to one and only one point in set $B$ . How must we connect to use shortest length of wire??
I know the answer is take the smallest element in $A$ and connect to smallest element in $B$ , then do so for second-smallest ones and so on.. I was trying to proof it . I tried using induction.. I also tried trying to decompose other possible connections to the one that is the solution.. but I failed.. miserably.