By Quote: all roads in set have not common ends Do you mean that no roads share a city? (So, you can't have a road going from $A$ to $B$, and a road going from $B$ to $C$) Or do you mean two roads don't go between the same pair of cities?

Between two cities can be only one road.In ideal set roads don`t share a city.

Suppose on the contrary, after $199$ days there is still a road between- say $A$ and $B$. Since all roads going to either $A$ or $B$, not including $AB$ itself, are at most $198$, there would exist a day when non of these roads is destroyed. But on that day we breach the ideal set definition, since we can add $AB$ to the destroyed roads on that day and they still would have not common ends. A contradiction. Comment. There is a way of destroying roads, such that all roads will be destroyed in no more than $101$ days. Indeed, the Vizing's theorem says we can color the edges of the corresponding graph with no more than $101$ colors. Thus, removing every day some of the colors, the result follows.

Induct on n and use Hall Marriage Theroem.